�����

����� ����� �������������� �"!$#�%'&"(*)+(-, .0/�(

1325476�894"8-47:<;�8-=>2@?BADC5=B:38$AE=FA+G34�H3ID=3JB2K4MLN8OCP:QA+G>CP8R4�;>C@ADCK=B:SJMTVUXWBYMZ\[F]3Z_^a`-bc3d�dBeMfhg i =BH�Tj=>kOlnm0oBpXqrlnm0oBsut3m$vwlnmyxzpOq<l|{�}�~�������CP2@2�J34VH<ID478-47:>AE4M;�AD==B:<4NH3ID4��E?<:BC��B47I�8*C�A�TSID476�;34MI�� G3=Q8-47:<;38OCK:S89=>2P?BADCK=B:38�J347ku=BID4"A+G34N;<476�;>2PCP:<4 f�� G34;<4 i CK8+C5=B:F=>k<A+G34 4M;BC�AE=BI�CK8��<:36M2 f� =B8��B?r478+ADCK=B:38�H3ID=M�>C547:3:<4M:>A i 4�L�=>CP8-� i CX;B?���=B: i =>?<I�8���������;<4��w6�A+G��4��LF6�ADC5�>?r4O;>?�8-4 i =B:<;B6MCPID4X;<4X��=>2K=BL�J>C54��rIDC�AD6M:3:BC5�>?r4 f3� =>?<8�ID4MLF4MI i CK=B:38�� 2@CK:>Az¡M4�4>¢£$¤ 6�:36�¥�6�:F��=>2@254M¥74B¢M¦�4MID:<=B:y¢�� ��¢3�B?3Cr8�§ = i+i ?<H<4 ;<4 i 478 i =B: i =>?<I�8 f

¨N©Rªa«�©X¬ ­<®°¯�±O±O²°³�´¶µ�·�¸�¹Fº´�»¶·�¸0¼+½X¬�´³O¬¾®�´$«�©�ª�³�·�¼E­r´°³a´¿¨N©OÀ+©�»ÂÁO¼+´ÂÃ�­r¼u¸�·$ª�ªO¼+½O¬ ´Ä ��Å�Æ�ÇÉÈ0ÊuÅ�Ë Ì�Ç�Í�ÎÇ Å�Ê���Ï"Ð�ËyÏ-ÑMÊ�ÇnÒ$Ó�Ô�Ç Å�Æ�Ï7ÇyÆ0Ê�ÕVÖ�ËrÊ��y�0�0×

Ø f0Ù �4\AE47IEL�CK:<47I�254a:<=BLFJ3ID4 ;<4a8��4M�B?r47: i 478�;y§ 4M:>ADCK4MI�8 i =B:38��4 i ?BADC@k58�;<=B:>A�2K6 89=BLFLF44M8EA100

f¯ fRÚ :�I��4M894MI��B=>CPIN�>C@AEI��4 i ?rJBCK�B?r4j;34 i7Û=�A7�4É?<:ÜL�Ý4\AEID4j4M8EAQH32P6 i �4É8*?<I�?<:<4�A�6�J>254G3=BIDC�Þ*=B:>AD67254�4\AX478+A$ID4MLNH32@C$Ý6�L�=>C�ADCD�4";y§ 4M67? f<g CK:38+C�¢<2K6"H<ID=>ku=B:<;347?<IX;34"2�§ 476M?V;B6�:38254�I��4M894MI��B=>CPIzßK2P6 ;BCP8+A�6�: i 4a4M:>AEID4R2P6a8*?<I�k56 i 4a;<4R2�§ 476M?�4\A�2P6a8+?<IDk56 i 4a;34�2P6RA�6�J>254_à�4M8EA?<:�;34ML�C5�+L�Ý4\AEID4 f ¡M4 I��478-47I��B=>CKI�4M8EA�AE=>?<ID:��4�6M?BAD=>?<I$; § ?<:<4";34M8�6MI Û4\AE478�8*?<Iz2K6aA�6�J>2546M�<:Q�>?�§ ?<:<4";34M8�k56 i 4M8�;B?QI��4M894MI��B=>CPI�67C@Az?<:F6�:<¥\2K4�;<4

30o 6\�B4 i 2P6aA�6�J>254 f<� ID=>?>�B47I2K6aH<ID=>ku=B:<;347?<I�;34�2�§ 4M67?F6�H3I�Ý478�2P6aID=�AD6�ADC5=B: fá f ¡M478�2K=B:<¥\?r47?<I�8O;<478 i7Û=�A��478O; § ?<:FAEIDCP6�:<¥\2K4�8-=B:>A

13¢134\A

10f ¡M4â+ã\ä*âæå�ã�çKè3é\âDä�çæê ;34 i 4$AEI�CK6M:<¥\254X4M8EA0?<: i 4MI i 2K4X6\T>6�:>A�89=B: i 47:>AEID4�Ý6X2�§ CP:<�AEIDCK47?<I�;>?OAEIDCP6�:<¥\2K4$�>?3C�478+A<AD6M:<¥74M:>A�Ý6 i GB6 i ?<:a;34M8 i7Û=�A7�4M80;B?OAEIDCP6�:<¥\2K4 fß@¦�=>CPI�2K4 ;>CK6�¥�I�6�LNL�4 f à � ID=>?>�B47Iz2K4�I�6�TB=B:F;>? i 4MI i 2K4�CP:38 i I�C@A f

.

................................................................................................................................................................................................................................................................................

.

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

.

.

..

.

.

..

.

..

..

.

..

.

..

..

..

..

..

..

..

..

..........................................................................

..

.

..

..

.

..

.

..

.

.

.

..

.

.

..

.

..

..

..

......................................................

....................................................

ë f � CP:<��47:>ADC547I�8$Hr=B8+C@ADC@k58a¢b¢c¢d4\A

e8+?<H��47IDCK47?<I�8 Ý6a?<:FID47LFHB2PCP89894M:>A�2K4M8 i =B:<;>C5�ADC5=B:38�8+?3C��>6M:>AE478�ì

a(b + c + d + e) = 128¢

b(a + c + d + e) = 155¢

c(a + b + d + e) = 203¢

d(a + b + c + e) = 243¢

e(a + b + c + d) = 275f

� ID=>?>�B4MI i 478 i CK:<�F47:>ADC547I�8 fí f�Ú :V6�IEJ3ID4�J>CK:367CKID4F4M:>ADCK4MI i =B:38+CK8EAE4F47:V?<:�î3}Bï\ð<ñ�I�6 i CP:<4��>?3C�6�;<4�?�òQï�î�ó�ô7îMõ*ö�¢?<:":<=347?r;";3ID=>C�A�4\A�?<:":<=347?r;"¥�6M? i G34>¢>4�A i GB6 i ?<:�;<478z:<=347?r;B8�4M:Bk56M:>AD8z6�;<4�?�òa4M:<�k56�:>A�8�¢M÷5?<8-�>?�§uÝ6 i 4a�B?r4R2K4�GB6M?BAz;<4R2�§ 6MI+JBID4�8-=>C�A�6�A�AE4�CK:>A-¢B=�Ý? i GB6��B?r4�:<=34�?r;�:y§ 6aH36�8;y§ 4M:Bk56M:>AD8 fyÙ 6�:38z?<: i 47I�A�6MCP:�6MI+JBID4aJ>CK:367CKID4�47:>ADC547I-¢ i GB6��>?r4R:<=347?r;�4M8EA�:B?<L��47ID=�A7�4B¢

�����

4M: i =BLFLF4M:��i 6�:>A�H36�I1Ý6X2K6XI�6 i CP:<4O4�A�4M: :B?<L��47ID=�AD6M:>A�;34�2K6O¥�6M? i G34��B4MI�8�2P6R;3ID=>C�AE4Ý6aAEI�6\�B4MI�8 i GB6��B?r4�:BC �B4M67? f ¡M4�;BCP6�¥�I�6MLFLF4�LF=B:>AEID4 2K4M8��B?<6�AEID4�H<ID47LNCK4MI�8�:BC��B476M?�ò;y§ ?<:�AD472�6MI+JBID4 f ¡76VI�6 i CK:<4V; § ?<:wAE4�2�6�IEJ3ID4V4M8EA�HB2K6 i �4jÝ6S2�§ =BIDCK¥\CK:<4V; § ?<:É8�T38EA7Ý4MLF4;<4 i =3=BID;<=B:3:��4M4M8

xy¢�2�§ 6_ò�4

x�4�A�6�:>A�Ý6Q2�§ G3=BI�C@Þ*=B:>A�6M2z4�AR2�§ 6\ò�4

y�4\AD6M:>A�Ý6�2K6N�B4MIæADC5�i 67254>¢ i =BLFLF4�C@2P2@?<8+AEI��4 f�� Cy2�§ 4789H36 i 4�47:>AEID4�2K4M8�:BC��B476M?�òN;<4 2�§ 6MI+JBID4�4M8EA$;<4

2?<:BC�A7�4M8;36M:38z2K6�;BCPID4 i ADCK=B:

y4�A�2�§ 4789H36 i 4MLF4M:>A�4M:>AEID4O2K4M8z:<=347?r;B8�;B?N:BC �B4M67?N8+?<H��47IDCK47?<I��>?3Ci =B:>ADCK4M:>A�2K4O:<=347?r;�:B?<L��47ID=�A7�4

20064M8EA�;34

2?<:BC�A7�4M8�;36M:38�2K6�;>CKID4 i ADC5=B:

x¢MAEID=>?>�B47I25478 i =3=BID;3=B:3:��4�478�;>?�:<=34�?r;N:B?<L��47ID=�A7�4

2006f

..

..

..

..

..

..

..

.....................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

.....................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

.....................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

.....................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

....................................................

.

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

................................................... .

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

................................................... .

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

................................................... .

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

...................................................

..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

.................................................... ..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

....................................................

.

..

..

..

..

..

..

..

..

................................

...

..

..

..

..

..

...................................................

............................................

............................................

............................................

............................................

............................................

............................................

............................................

............................................

.................................................................

.................................................................

.................................................................

.................................................................

...........................................................................................................................

...........................................................................................................................

........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........

.

.

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

.

.

..

..

.

..

..

.

..

.

..

.

.

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

.

.

..

..

.

..

..

.

..

.

..

.

.

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

.

-

6

x

y

8 9 10 11 12 13 14 15

4 5 6 7

2 3

1

Ã"­r¼u¸�¼E®�¹ ¨N©OÀ�¬ »ÂÁO¼E·���´�«�©Rªa³R·$­���«�¹R©a©OÀµ�·�¸�¹R´$»¶·�¸�¼+«�® ¨N©�ª�¸0´$® ¸ ¯�±O±O²ÍrÇ Å�Ê���ÏQÈ�ÊuÅ�ËrÌ Ä ����Å�ÆzÓ� �Ë Ï-Ñ�Ò$ÓzÈ�Ï�Ê�Æ�Ë��zÓ�� Ë��hÕ�Ó��y�0�0×

Ø fFÙ 4\AE47IEL�CK:<4 A+G34�:B?<LFJ<47IO=>k�894��>?r4M: i 4M8R=>k i =B:3894 i ?BADC �B4"CP:>AE4M¥74MI�8$� G3=B8-4"8*?<LCK8100

f¯ f gQi ?rJBC i 672�¥\2K6M898 A�6�: ¤ ��C�A+GX8+C5;34M80=>kM2K4M:<¥_A+GO=B:<4�LF4�AEID4zCP8�H32P6 i 4�;R=B:O6�G3=BIDC�Þ*=B:>AD672AD6�JB2K4X6�:<;�GB672PkB�<2@254M;R��C�A+GR�a6�AE47I f0� G>?<8�¢7A+G34O;34MH>A+G�=>kBA+G34��a6�AE47I0CP:aA+G34$A�6�: ¤ ß@A+G34;BCP8+A�6�: i 4 =>krA+G34a8+?<IDk56 i 4a=>krA+G34O�a67AD4MI�k5ID=BL A+G34�8*?<I�k56 i 4 =>krA+G34RA�6�J>254_à�CP8�=B:<4aGB6M2@kL�4\AEID4 f0� G34$A�6�: ¤ CK8�ID=�AD6�AE4M; 6�J3=>?BA�=B:<4O=>k>A+G34O4�;3¥74M8�A+GB6�A�CP8z=B:�A+G34$AD6�JB2K4X8-=RA+GB6�A=B:<4Rk56 i 4a=>k<A+G34OA�6�: ¤ LN6 ¤ 478�6

30◦ 6�:<¥\2K4X��C�A+G A+G34OAD6�JB2K4 f�� CK:<; A+G34a;<47HBA+GN=>k<A+G34�a67AD4MI�CK:"A+G34OAD6M: ¤ 6MkPAD4MI�A+G34aID=�A�67ADCK=B: fá f0� G34X2K4M:<¥_A+GB8�=>kBA+G34O8+C5;34M8z=>k<6XAEIDCP6�:<¥\2K4O6MID4

13¢13¢>6M:<;

10f�� G34ç5èBé�âDä�ç��3ã��aâæçKä*âæå�ã =>k�A+G>CK8�AEIDCP6�:<¥\2K4�CP8�A+G34 i CKI i 2K4z��C�A+G i 47:>AEID4�CK:38+C5;34�A+G34i CKI i 2K4 A+GB67A�CK8�A�6�:<¥747:>A�AE=Q4M6 i G�=>k�A+G34�A+GBID4M4�8+C5;34M8O=>k0A+G34 AEI�CK6M:<¥\254 fß � 4M4RA+G34 ;>CK6�¥�I�6�L f à � CK:<;�A+G34aI�6�;BC@?<8�=>krA+G34RCK:38 i IDCKJ<4M; i CPI i 254 f

.................................................................................................................................................................................................................................................................................

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

.

.

..

.

.

..

..

..

..

.

..

..

..

..

..

..

............................................................................

..

..

..

..

.

..

.

.

..

.

.

.

..

.

.

..

..

..

.

.......................................................

....................................................

ë f�� C �B4$H<=B8*C�ADC��B4�CP:>AE4M¥74MI�8a¢b¢c¢d¢76�:<;

e¥�ID4767AD4MI A+GB6M:a=B:<4$LN6 ¤ 4�A+G34�ku=>2P2K=���CK:<¥i =B:<;>C@ADCK=B:38�AEI�?r43ì

a(b + c + d + e) = 128¢

b(a + c + d + e) = 155¢

c(a + b + d + e) = 203¢

d(a + b + c + e) = 243¢

e(a + b + c + d) = 275f

� CP:<;"A+G34��>�B4�CP:>AE4M¥74MI�8 f

�����

í f�g kK?32@20JBCP:36�I�TNAEID4�4 i =B:38+CK8EAD8X=>k06"ID=3=�A�î3}>ñBï�� G>C i GQGB6M8$Aæ� =�������~@ñ��*ï�îy¢r6"I�C5¥7G�Ai G>CP2K;N:<=3;<4a6M:<;N6a2K47kPA i G>C@25;�:<=3;<4>¢36M:<;F4M6 i G i G>C@25;�:<=3;<4 GB6M8�Aæ� = i G>CP2K;3ID47:y¢B?<:>ADCP2A+G34RAD=BHF=>krA+G34OAEID4M4�CP8�ID476 i G34M;y¢�� G347ID4 476 i G�:<=3;34aGB6�8�:<= i G>CP2K;3ID47: f� :N6 i 4MIæAD67CK:kK?32P2�JBCP:36�I�TXAEID4�4�4M6 i GO:<=3;<4�CP80:B?<L�J34MID4M;y¢78+A�6�IæADCK:<¥���C@A+G167ArA+G34�ID=3=�A-¢7:B?<LFJ<47IDCP:<¥k5ID=BL 2K47kPA0AD= I�C5¥7G�A�6 i ID=B898z476 i Ga2K4-�B472 f0� G34O;BCP6�¥�I�6ML 89G3=��a8�A+G34X�<I�8+A0ku=>?<I�2K4-�B472P8z=>k8*? i GF6aAEID4�4 f�� G34�ID=3=�A$=>k A+G34aAEID4M4 CP8�HB2K6 i 4M;�6�AzA+G34�=BIDCK¥\CK:Q=>k�6�:

xy i =3=BID;BCP:367AD48�T38EAE47LS¢B��C�A+GNA+G34x 6\òMCK8XG3=BIDC�Þ*=B:>AD672�6�:<;NA+G34 y 6_ò�CP8��B47I�ADC i 672�¢r6M8X8-G3=��a: f�� k�A+G3489H36 i CP:<¥�J<4\Aæ� 4�47:QA+G34�2K4-�B472P8�=>k�A+G34"AEID4M4�CP8

2?<:BC�AD8OCP:QA+G34

y ;BCPID4 i ADCK=B:S6�:<;QA+G3489H36 i CP:<¥OJ34�Aæ� 4M4M: A+G34O:<=3;34M8�67A0A+G34�AE=BH�2K4-�B472�A+GB67A i =B:>A�6MCP:38�A+G34O:<=3;34R:B?<LFJ<47ID4�;2006

CP82?<:BC@A�8�CP:aA+G34

x ;BCPID4 i ADCK=B:y¢��<:<;�A+G34 i =3=BID;BCP:367AD4M8�=>k>A+G34X:<=3;<4X:B?<LFJ<47ID4�;2006

f....................................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

..

.....................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

.......................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

..

.....................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

.......................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

..

.....................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

.......................................

..

..

.

..

..

.................................................... .

..

..

..

..

..

..

..

.....................................

..

..

.

..

..

....................................................

.

.

..

..

.

..

..

..

..

...................................

...

..

..

.

..

...

.................................................. .

.

..

..

.

..

..

..

..

...................................

...

..

..

.

..

...

.................................................. .

.

..

..

.

..

..

..

..

...................................

...

..

..

.

..

...

.................................................. .

.

..

..

.

..

..

..

..

...................................

...

..

..

.

..

...

..................................................

.

..

..

.

..

..

..

..

..

...................................

...

..

.

..

..

.................................................... .

..

..

.

..

..

..

..

..

...................................

...

..

.

..

..

....................................................

.

.

..

..

.

..

..

..

..

..

.................................

...

..

.

..

..

...

..................................................

............................................

............................................

............................................

............................................

............................................

............................................

............................................

............................................

.................................................................

.................................................................

.................................................................

.................................................................

...........................................................................................................................

...........................................................................................................................

........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........

..

.

..

.

..

..

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

.

.

..

.

..

..

..

.

..

.

..

..

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

.

.

..

.

..

.

.

..

.

..

.

..

..

..

.

..

.

.

.

..

.

..

.

-

6

x

y

8 9 10 11 12 13 14 15

4 5 6 7

2 3

1

� 4+òMA�� 4Ü¥\C �B4nA+G34h89=>2P?BADCK=B:38�AE=�A+G34 � 67ADCK=B:36M2Q�r6M: ¤ =>k � 4_� �74M672K6M:<;��=BLNHr4\ADC@ADCK=B:������������������Fì �<��� � � ��� fØ f�� �+ô�ñBï��a}�î>~�� f�� :"A+G>CP8�H3ID=3JB2K4ML � 4>§ 2@2 J<4�HB2K6 i CK:<¥$�>6�I�C5=>?<86�IDI�6�:<¥747L�47:>AD8N=>k10i| 6M:<; 20

i| i =>CK:38�=B:�A+G34�:BCK:<4�8-�>?<6�ID478=>kz63 × 3

¥�IDCK; f�� òM6 i AD2 T�=B:<4 i =>CK:���C@2P2�J34�HB2K6 i 4M;VCK:�476 i G=>k0A+G34�:BCK:<4�89�B?<6MID4M8 fa� G34N¥�I�C5;SGB6M8Oku=>?<I2 × 2

8*?rJB8-�>?<6�ID4784M6 i G i =B:>A�6MCP:BCK:<¥X6 i =BID:<4MI9¢�A+G34 i 4M:>AEID4>¢>6�:<; A+G34XAæ� =�8-�>?<6�ID4786�;�÷u6 i 4M:>A�AE=�A+G3478-4 f�£ :<4 4+òM6�LNH32K4�CP8�8-G3=��a:�CP:"A+G34 ;>CK6�¥�I�6�L f....................................................................................................................................................................................................................

....................................................................................................................................................................................................................

....................................................................................................................................................................................................................

....................................................................................................................................................................................................................

.

..

.

..

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

....................................................10 .

.

.

.

.

.

.

..

.

..

.

..

..

...........................................

.......................................................

..............................20 .

.

.

.

.

.

.

..

.

..

.

..

..

...........................................

...

.

..

.

.

.

.

.

.

.

.

.

.

..

.

.

.

...

.............................................................20

.

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

...................................................10 .

..

..

..

..

..

..

....................................

...

..

..

..

..

..

...................................................10 .

..

..

..

..

..

..

..

..................................

...

..

..

..

..

..

...................................................10

..

..

..

..

..

..

..

..

...................................

..

..

..

..

..

....................................................10 ..

..

..

..

..

..

..

.....................................

..

..

..

..

..

....................................................10 .

.

.

.

.

.

.

..

.

..

.

..

..

...........................................

...

.

..

.

.

.

.

.

.

.

.

.

.

..

.

.

.

...

.............................................................20

ßK6�à � CP:<;�6�:N6MIEI�6M:<¥74MLF4M:>A�� G347ID4RA+G34OAE=�A�6M2P8$=>k<A+G34�ku=>?<I2 × 2

8*?rJB8-�>?<6�ID478$6�ID440i| ¢ 60 i| ¢ 60 i| ¢�6�:<; 70

i| CP:w6�:�Tw=BID;<47I f ß Ù I�6��¶6V;>CK6�¥�I�6�L 8-G3=���CP:<¥�TB=>?<I6�IDI�6�:<¥747L�47:>A f àßuJ�à � CP:<;�6�:N6MIEI�6M:<¥74MLF4M:>A�� G347ID4RA+G34OAE=�A�6M2P8$=>k<A+G34�ku=>?<I

2 × 28*?rJB8-�>?<6�ID478$6�ID4

50i| ¢ 60 i| ¢ 70 i| ¢�6�:<; 80

i| CP:w6�:�Tw=BID;<47I f ß Ù I�6��¶6V;>CK6�¥�I�6�L 8-G3=���CP:<¥�TB=>?<I6�IDI�6�:<¥747L�47:>A f à� =BI�4M6 i G"H<6MI�Az=>k3A+G34�H3ID=3JB2K4ML°J<4�25=���¢�CP2@2P?<8EAEI�67AD4�TB=>?<I�6�:38D� 4MI0��C�A+G"6�8*?3C�A+�6�J>254�6�IDI�6�:<¥747L�47:>A�6M:<; 6�:�4*ò�H32P6�:36�ADC5=B:�=>k>� G�T :<= =�A+G34MI�8+?3C@A�6�J>254�6�IDI�6�:<¥74M�L�47:>A i =B:>A�6MCP:38�6�2P6�ID¥747I�ß5H36�IæA�ß i à5àr=BIy6�89LN6M2@2547Iyß5H36�IæA�ßu;�àuà<6�LF=>?<:>A�=>k�L�=B:<4-T f

ß i à��SGB67AyCK8�A+G34�LN6\òMCKL�?<L 6�LF=>?<:>A�=>kBLF=B:<4-TX� G>C i G i 6�: J34�HB2K6 i 4M;a=B:RA+G34�¥�IDCK;8-= A+GB6�A�4M6 i GF=>k<A+G342 × 2

8*?rJB8-�>?<6�ID478 i =B:>AD67CK:38$4*òM6 i AD2�T50i| �

��� �

ßu;�à �SGB67A�CK8 A+G34SL�CK:BCPLN?<L 6�LF=>?<:>A�=>k$L�=B:<4-TV� G>C i G i 6M:ÉJ<4�H32P6 i 4�;É=B:�A+G34¥�IDCK;�8-=FA+GB6�A�A+G34N6\�B4MI�6�¥74N=>k0A+G34�6�LF=>?<:>AO=>k�L�=B:<4-TQCP:V476 i G�=>k0A+G342 × 28*?rJB8-�>?<6�ID478�CP8$4*òM6 i AD2�T

60i| �

��� å��3êKç � è ���Nê� ã ã���ç�ê � ä fß56\à � CP: i 4 =B:<4

2 × 28*?rJB8-�>?<6�ID4 i =B:>A�6MCP:38�=B:B2�T

40i| ¢B6M2@2y=>k C@A�8�4M:>AEI�C5478$LN?<8EAJ<4

10i| i =>CK:38 f�� :FH36�IæADC i ?32P6�I9¢>A+G34 i 47:>AEID4 89�B?<6MID4aL�?<8+A i =B:>A�6MCP:N6 10

i| i =>CK: f�� CK: i 46�:<=�A+G347I�=>k>A+G342×2

8+?rJ389�B?<6MID4M8 i =B:>AD67CK:3870i| 6�:<;�C@A�L�?<8+A�672K89= i =B:>AD67CK:�A+G34 10

i|i =>CP:FCK:NA+G34 i 47:>AEID4�8-�>?<6�ID4>¢<C@A�8�ID4MLN6MCP:BCK:<¥ 4M:>AEI�C5478�LN?<8EA$J<420i| f � A�CP8�A+G34M:Q476�8�TAE= i =BLNH32K4�AD4aA+G34�89�B?<6MID4"6M8�CP: � C5¥\?<ID4 � J3472K=�� fX£ A+G34MI�89�B?<6MID4M8X6MID4�Hr=B8*8*CKJB2K4�JMTID=�AD6�ADC5=B: f

ßuJ�à � CP: i 4X=B:<42×2

8*?rJB8-�>?<6�ID4 i =B:>AD67CK:3880i| ¢�672P2>=>kBC�AD8�4M:>AEI�C5478�LN?<8EA�J34 20

i|i =>CP:38 f � : H36�IæADC i ?32P6�I9¢7A+G34 i 4M:>AEID4�89�B?<6MID4�LN?<8EA i =B:>AD67CK:a620i| i =>CK: f � CK: i 4�6�:<=�A+G347I=>k�A+G34

2 × 28*?rJB8-�>?<6�ID478 i =B:>AD67CK:38R=B:B2�T

50i| 6�:<;QC@A�L�?<8+A�672K89= i =B:>A�6MCP:�A+G34 20

i|i =>CP:FCK:NA+G34 i 47:>AEID4�8-�>?<6�ID4>¢<C@A�8�ID4MLN6MCP:BCK:<¥ 4M:>AEI�C5478�LN?<8EA$J<410i| f � A�CP8�A+G34M:Q476�8�TAE= i =BLNH32K4�AD4aA+G34�89�B?<6MID4"6M8�CP: � C5¥\?<ID4��"J3472K=�� fX£ A+G34MI�89�B?<6MID4M8X6MID4�Hr=B8*8*CKJB2K4�JMTID=�AD6�ADC5=B: f

ß i à � G34N6�LF=>?<:>AR=>k�L�=B:<4-T���CP2@2�J<4NLF6_ò�CPLNC�Þ*4�;F� G34M:QA+G34F:B?<LFJ<47IR=>k20i|i =>CP:38R=B:FA+G34�¥�IDCK;QCP8XLF6_ò�CPLNC�Þ*4�; f � k�� 4"HB2K6 i 4�6

20i| i =>CP:QCP:�A+G34 i 4M:>AEID4"89�B?<6MID4B¢A+G34M: 672P2MA+G34�ID4MLN6MCP:BCK:<¥ i =>CK:38�GB6��B4�AD=aJ<4

10i| i =>CP:38 f � k�� 4$H32P6 i 4�6 20

i| i =>CK:a6M2K=B:<¥=B:<4�8*CK;<4�=>k>A+G34X¥�I�C5; ¢MCP:�A+G34�L�C5;3;B2K4�8-�>?<6�ID4X=>k�A+GB67A08*CK;<4>¢�A+G34M:a67A0L�=B8EAyAæ� =�L�=BID420i| i =>CP:38�LF6\T J34�H32P6 i 4�;a=B:RA+G34X¥�IDCK; ßPCK:RA+G34 i =BIE:<47I�8�=B:RA+G34X=BH3Hr=B8+C@AD4$8*CK;<4_à f �<?BA� 4 i 6M:�?<8-4Rku=>?<I

20i| i =>CP:38�CPk<A+G34-TN6MID4�H32P6 i 4�;"CK:�A+G34Rku=>?<I i =BIE:<47I�8$=>krA+G34a¥�IDCK;N6M8CK: � C5¥\?<ID4 � J3472K=�� f0� G>?<8\¢�A+G34OLF6_ò�CPLN?<L :B?<L�J34MI�=>k

20i| i =>CP:38�A+GB67A i 6M:NJ34X?<8-4M;CK8�ku=>?<I9¢<¥\C �BCP:<¥R6�AD=�AD672 =>k

$1.30f

ßu;�à � G34 ;BCP8+AEI�C5J>?BADC5=B:�CP: � CK¥\?<ID4 � J<4�25=�� 8*67ADCP8*�r4M8�A+G34 i =B:<;BC�ADC5=B:38�6�:<;�?<894M8$1.30

¢\� G>C i GOCP: i 2P?r;34M8�=B:B2�TXA+GBID4�420i| i =>CK:38 f � kM� 4$6MID4�AD=O25=�� 47IrA+GB67A AE=�A�6M2�6�A�6M2@2�¢� 4 LN?<8EA�?<8-4 6�A�LF=B8+A�Aæ� =

20i| i =>CK:38 f �<?BAzA+G347:�:<=B:<4"=>k A+G34 2 × 2

8*?rJB8-�>?<6�ID478i =>?32K; i =B:>AD67CK:�L�=BID4QA+GB6M:wAæ� =20i| i =>CK:38\¢�6�:<;�A+G34MID4�ku=BID4V:<=B:<4 i =>?325; i =B:>A�6MCP:L�=BID4 A+GB6M:

60i| f � CP: i 4 A+G34"6��B47I�6�¥74"6 i ID=B8*8O672P2yku=>?<I 2 × 2

8*?rJB8-�>?<6�ID478OLN?<8EAXJ<460i| ¢04M6 i GS=>k�A+G34�ku=>?<I�� =>?325;�A+G347:�GB6\�B4"AD= i =B:>A�6MCP:V4*òM6 i AD2�T 60

i| ¢r� G>C i G�L�476�:38A+GB67A�4M6 i G$� =>?325;RGB6��B4�AE= i =B:>AD67CK:XAæ� =20i| i =>CP:38 f�� G34�8*6�LF4zAæ� = 20

i| i =>CP:38 � =>?325;A+G34M:�GB6\�B4�AD=FJ34�CK:N6M2@2rku=>?<I2 × 2

8+?rJ389�B?<6MID4M8\¢�� G>C i G�CP8�CPLFH<=B898+C5J>254 f�� G>?<8\¢BA+G34LNCP:BCKL�?<L°CK8$1.30

f

...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 .

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 ..

..

..

..

.

..

..

..

..

...............................................

...

..

..

..

..

.

...

..................................................................20

.

.

.

.

.

..

..

..

...............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

..

..

.

..

..

..

..

..

..............................................

...

...

..

.

..

..

..

...................................................................20

.

..

..

..

.

..

..

..

..

..

...............................................

...

..

..

..

.

...

..

.................................................................20 .

..

..

..

.

..

..

..

..

..

..

.............................................

...

..

..

..

.

...

..

.................................................................20 .

..

..

..

.

..

..

..

..

..

..

.............................................

...

..

..

..

.

...

..

.................................................................20

� C5¥\?<ID4 � ...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 .

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 .

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10

.

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

..

..

.

..

..

..

..

..

..

..

..........................................

...

...

..

.

..

..

..

...................................................................20 .

.

..

..

.

..

..

..

..

..

..............................................

...

...

..

.

..

..

..

...................................................................20

.

..

..

..

.

..

..

..

..

..

...............................................

...

..

..

..

.

...

..

.................................................................20 .

..

..

..

.

..

..

..

..

..

..

.............................................

...

..

..

..

.

...

..

.................................................................20 .

..

..

..

.

..

..

..

..

..

...............................................

...

..

..

..

.

...

..

.................................................................20

� CK¥\?<ID4�� ...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

..

..

..

..

.

..

..

..

..

..

.............................................

...

..

..

..

..

.

...

..................................................................20 .

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 ..

..

..

..

.

..

..

..

..

..

.............................................

...

..

..

..

..

.

...

..................................................................20

.

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

.

.

.

..

..

..

...............................

..

.

.

.

.

.

..

.

.

.

...........................................10

.

..

..

..

.

..

..

..

..

..

..

.............................................

...

..

..

..

.

...

..

.................................................................20 .

.

.

.

..

..

..

..

.............................

..

..

.

.

.

.

.

.

..

.

...

........................................10 .

..

..

..

.

..

..

..

..

..

...............................................

...

..

..

..

.

...

..

.................................................................20

� CK¥\?<ID4 � ...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

...................................................................................................................................................................................................

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10 ..

..

..

..

.

..

..

..

..

...............................................

...

..

..

..

..

.

...

..................................................................20 .

.

.

..

.

..

..

..

...............................

.

..

.

.

.

.

.

.

.

...

.........................................10

.

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10 .

.

..

..

.

..

..

..

..

..

..............................................

...

...

..

.

..

..

..

...................................................................20 .

.

.

.

.

..

..

..

..

.............................

..

.

.

.

.

.

..

.

.

.

...........................................10

.

.

.

.

..

..

..

..

.............................

..

..

.

.

.

.

.

.

..

.

...

........................................10 .

..

..

..

.

..

..

..

..

..

...............................................

...

..

..

..

.

...

..

.................................................................20 .

.

.

.

..

..

..

..

.............................

..

..

.

.

.

.

.

.

..

.

........................

...................10

� C5¥\?<ID4 �

��� �

¯ f�� ?<LF6M: ¤ CP:<;w�a6M8�ID4 i 47:>AD2�T i =B:>AD6 i AE4M;�JMT�A+GBID4�4S6M2@C547:ÉI�6 i 4M8�ìNA+G34�� � 4M4M:38\¢A+G34 £ Þ*;36 ¤ 8\¢B6�:<;"A+G34���47I ¤ ?<:38 f ¡�C�A�AD254RCP8 ¤ :<=��a:F6�J<=>?BA�A+G34M894aI�6 i 478�4*ò i 4MH>A�A+GB6�A� � 4M4M:38�672��a6\T38�89H<4M6 ¤ A+G34QAEI�?BA+G�� G>C@254 £ Þ*;B6 ¤ 8"6M2 �a6�TB8�2PCK4 f � :�6�:�T�¥�ID=>?<H�=>k6M2@C547:38�6���4MI ¤ ?<:���CP2@2<:<4-�B4MIz8*Hr476 ¤ �<I�8+A f �SG34M:�C�A�;3=34M8�89H<4M6 ¤ ¢>C�A�AE4�2P2P8zA+G34OAEID?BA+GCPk�A+G34�H3ID4-�BCK=>?<8a8+A�67AD4MLF4M:>A$�a6�8a6F2@C54>¢�6M:<;V2@C5478�C@k�A+G34FH3ID4-�BCK=>?<8 8EAD6�AE47L�47:>A��a6M8AEID?BA+G>kK?32 f0g 2@A+G3=>?r¥7G�A+G34�6M2@C547:38 i 6M:NID4M6�;BC@2�T�AE4�2P2 =B:<4�6�:<=�A+G347Iz6MH<6MI�A-¢3=>k i =>?<I�894RAD=G>?<LF6M:38$6M2@2r6M2@C547:38�25=3= ¤ A+G34�896ML�4 fg G>C5¥7G3�E254-�B4�2y;3472K4�¥�6�ADC5=B:F=>krA+GBID4M4a6M2@C547:38$GB6�8�J34�47:F894M:>A�AD= � 6�IæA+G"AD=�:<4�¥7=3�ADCK6�AE4a=>?<I�k56�AE4 f�g L�=B:<¥�A+G34ML CK8�67A�25476�8EAz=B:<4�� � 4M4M: f�£ :N6MIEI�C��>6723A+G34-T�LF6 ¤ 4RA+G34ku=>2P2K=���CK:<¥O8+A�67AD4MLF4M:>A�8�ßPCK:F=BID;34MI�à*ì

� CPI�8+A g 2PCK4M:�ì � G34�8-4 i =B:<;N672PCK4M:"CK8�6���4MI ¤ ?<: f� 4 i =B:<; g 2@C547:�ì � G34OA+G>CKID;N672PCK4M:"CK8�:<=�Az6���4MI ¤ ?<: f� G>CPID; g 2PCK4M:�ì � G34R�rI�8EAz6M2@C547:�CP8�6N��47I ¤ ?<: f

�SG>C i G�6M2@C547:F=BI�6M2@C547:38 i 6M:�TB=>?�J<4 i 47I�A�6MCP:F6MID4��r� 4�47:38 ���� âæç�å é � å��3êKç � è���ã� ���Mè �<ã � ���Nê�yãaã���ç�ê � ä f

� G34��rI�8EA�672PCK4M: i 6M:3:<=�A�J<4a6N��4MI ¤ ?<:y¢38*CP: i 4a6N��47I ¤ ?<:�:<4-�B47I�89H<4M6 ¤ 8��rI�8EA f� G>?<8\¢�A+G34FA+G>CPID;�8+A�67AD4MLF4M:>A�LN?<8EAaJ<4Q6�2PCK4B¢�� G>C i G�L�476�:38�A+GB67AOA+G34FA+G>CPID;�6M2@C547:i 6M:3:<=�A�J34a6�� � 4M4M: f� ?<H<H<=B8-4�A+GB67A0A+G34�A+G>CPID;"672PCK4M: CP8z6 ��4MI ¤ ?<:�ß@� G3=aCP8�2 TBCP:<¥-à f � CK: i 4���47I ¤ ?<:38=B:B2�T�2PCK4O� G347:NA+G34aH3ID4-�BCK=>?<8�8EAD6�AE47L�47:>AzCK8zAEID?r4>¢BA+G34a8-4 i =B:<;N8EAD6�AE47L�47:>A�LN?<8EA�J<4AEID?r4 f �<?BA�A+G347:"� 4aGB6\�B4a6 i =B:>AEI�6�;BC i ADC5=B: f�� 47: i 4>¢�A+G34RA+G>CKID;�6M2@C547:�CK8�6M: £ Þ*;36 ¤ f� CK: i 4zA+G34MID4�CK806��r� 4�47:�6ML�=B:<¥�A+G34zA+GBID4�4�672PCK4M:38\¢\C@AyL�?<8+A�J<4$=B:<4�=>kMA+G34��rI�8EAAæ� =N672PCK4M:38 f � ?<H<H<=B8-4RA+GB67A�A+G34a�<I�8+A�CK8$6�� � 4M4M: f�� G34M:�A+G34 8-4 i =B:<;�CK8$6N��47I ¤ ?<:6�:<;FC�AD8�8EAD6�AE47L�47:>A�CK8�6 2PCK4 f �<?BAzA+G>CP8�CPLFHB2PCK4M8�A+GB6�AzA+G34aA+G>CPID;�6M2@C547:FCP8�6���47I ¤ ?<:y¢� G>C i G�� 4�GB6��B4"672KID476�;�TQID?32K4�;�=>?BA fO� G347ID47ku=BID4>¢rA+G34��rI�8EA$6M2@C547:�CK8X:<=�A�6�� � 4M4M:y¢� G>C i G�L�476�:38zA+GB6�A�A+G34��rI�8EAz672PCK4M:�CP8�6�: £ Þ*;B6 ¤ 6�:<;�A+G34a894 i =B:<;�CP8�6��r� 4�47: f� G>?<8\¢7=B:B2�T$A+G34�8-4 i =B:<;X6M2@C547:XCK8�6��r� 4�47:�ßK6�:<;�A+G34�=�A+G347I�806MID4�J<=�A+G £ Þ*;B6 ¤ 8Eà f

á f ß � =�AE4Bì � :�A+G>CK8��B?r478+ADCK=B:N6M:��_4M�B?<672 ;>C��>CK8+C5=B:��$CP8�=B:<4X� G34MID4XA+G34OAE=�A�6M2>� 47CK¥7G�A=>krA+G34RAæ� ="H<6MI�A�8�CP8zA+G34a8*6�LF4 f àßK6�à�� 472@CK:<;B6�6�:<;R�yGB6�I�2547806MID4�J>?<ID¥\2K6MI�8 f g LF=B:<¥�A+G34�25=3=�A<k5ID=BLhA+G34�CKI<2K6�AE478+A i 6MHr47ICK8z6R8-4\A�=>k

12¥7=>25;�� 47CK¥7G�AD8�=>k

1¥�¢

2¥�¢

3¥�¢>6M:<;"8-=�=B:y¢�A+GBID=>?r¥7GaAE=

12¥ f � 6�:A+G34-TN;>C��>C5;34XA+G34X� 4�C5¥7G�A�8�4��>?<6M2@2�T�J<4\Aæ� 4�47:�A+G34ML � � k 8-= ¢>4*ò�H32P6MCP:NG3=��hA+G34-Ti 6M:F;<="C�A-¢B6M:<;�CPk :<=�A-¢�� G�TN:<=�A �ßuJ�à �SG34M:�� 472@CK:<;B6R6M:<;��yGB6�I�25478�AD6 ¤ 4�A+G34OID4MLN6MCP:<;<47I�=>kBA+G34X25=3=�A�AE= � ID4M;<;�T A+G34ku4M: i 4B¢zG34�;<47LF6M:<;38aA+G34

12¥�� 4�C5¥7G�Aa6M8"G>CP8�H<6\T3LF4M:>A f � 6�:n� 4�2PCP:<;36�6M:<;��GB6MID2K4M8X;>C��>C5;34aA+G34 ID47LF67CK:BCP:<¥

11� 4�C5¥7G�A�8X4��>?<6M2@2�T�J<4\Aæ� 4�47:NA+G34ML � � k08-= ¢4*ò�H32P6MCP:FG3=���A+G34-T i 6M:�;3="C@A-¢>6�:<;�C@ky:<=�A-¢�� G�T�:<=�A �ß i à�� 472@CK:<;B6 6�:<;��yGB6�I�25478�672K89=FGB6\�B4 6 8-4\A�=>k

1508*C@2��B47I�� 47CK¥7G�AD8�=>k

1¥�¢

2¥�¢

3¥�¢6�:<;É8-=É=B:y¢�A+GBID=>?r¥7G�AD=

150¥ f �y6M:�A+G34-Tj;BC �BCK;<4�A+G34M894Q� 47CK¥7G�AD8F4��>?<6M2@2�TJ<4\Aæ� 4�47:"A+G34ML � � ky89= ¢34+ò�HB2K67CK:FG3=���A+G34-T i 6M:F;<="C�A-¢B6�:<;"CPky:<=�A-¢�� G�T�:<=�A �

�����

��� âæç�å é � å��3êKç � è fß56\à���478�¢BC@A i 6M:QJ<4�;<=B:<4 f�£ :<4R�a6�TNCP8�AE=NH<67CKI�A+G347L k5ID=BL°A+G34 �_4+òMAEID47L�478 ��ì

1 + 12¢2 + 11

¢. . .¢6 + 7

f�� 6 i G�Hr47I�8-=B:"A+G347:"AD6 ¤ 4M8zA+GBID4�4a=>krA+G34�H<67CKI�8 f�� G347ID46�ID4a=�A+G34MI�Hr=B8+C5J>CP2@C@ADCK4M8 fßuJ�à���4M8\¢�C�A i 6�:�8EADCP2@2rJ34a;3=B:<4 f � 4�2PCP:<;36a¥74\AD8

6 + 16�:<;���GB6MID2K4M8�¥74�A�8

7f�� G34ID4M8EA i 6�:QJ34 H<67CKID4M;Fk5ID=BL°A+G34��_4+òMAEID47L�478 �Rß

2 + 11¢3 + 10

¢4 + 9

¢5 + 8

à+¢>��C�A+G4M6 i GNH<4MI�89=B:"AD6 ¤ CP:<¥XAæ� =N=>krA+G34�H<67CKI�8 fß i à � = ¢MC�A i 6�:3:<=�A�J<4X;<=B:<4 f0� G347ID4X6�ID4

754-�B4M: 6�:<;

75=3;3;R� 4�C5¥7G�A�8�¢�� G>C i GCKLNH32@C5478zA+GB67A�A+G34OAE=�A�6M2<CK8$=3;3;N6M:<; i 6M:3:<=�A�J<4�89HB2PC�A�4M�B?<672P2 T f

ë f�g�i G34M8*8-J3=B6�ID;SCP8a6M:8 × 8

¥�I�C5;�=>k�89�B?<6MID4M8 fQ£ :<4�=>k�A+G34 i G34M8*8aHBC54 i 478�¢�A+G34¤ CP:<¥�¢3LF=M�B478$=B:<4a89�B?<6MID4a67A�6�ADCPL�4RCK:�6�:�TF;>CKID4 i ADC5=B:y¢>CP: i 2P?r;>CK:<¥�;BCP6�¥7=B:3672P2 T fßK6�à g�¤ CK:<¥�ß5;<47:<=�AE4M;FJMT

KCK:"A+G34a;BCP6�¥�I�6ML"à�8EAD6M:<;38=B:�A+G34F2K=�� 4MI�254�kPA i =BID:<4MIR=>k�6 i G34M8*8-J3=B6�ID; f � AGB6�8�AD=QID4M6 i G�A+G34"8-�>?<6�ID4�LN6�I ¤ 4�;

FCP:S4*òM6 i AD2�TA+GBID4�4 L�=7�B4M8 f � G3=�� A+GB6�AzA+G34 ¤ CK:<¥ i 6M:Q;<="A+G>CK8CK:N4*òM6 i AD2�T"ku=>?<I�;>C���4MID47:>A��a6\T38 f

ßuJ�à g 898+?<L�4FA+GB67ARA+G34 ¤ CK:<¥�CP8"HB2K6 i 4M;ÉJB6 i*¤ =B:�A+G34J<=�A�AD=BL 254�kPA i =BID:<4MI f � :ÉG3=��¶LN6�:�TV�a6\T38 i 6M:C@AXID476 i GQA+G34�?<H3Hr47IX254�kPA i =BIE:<47IRßKLF6MI ¤ 4M;Gà�CP:4*òM6 i AD2�T�894-�B4M:NLF=M�B478 � ...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

K

F

G

��� âæç�å é � å��3êKç � è fß56\à � CP: i 4F476 i GSLF=M�B4NL�?<8+A��r:<;QA+G34 ¤ CP:<¥N=B:<4NID=��¿G>C5¥7G347I-¢yA+G34�ku=>2P2K=���CK:<¥ku=>?<I�;BCP6�¥�I�6MLF8�CP2@2P?<8EAEI�67AD4�6M2@2rHr=B8*8*CKJB2K4�ID=>?BAD4M8 f

................................................................................... ................................................................................... ................................................................................... ...................................................................................

................................................................................... ................................................................................... ................................................................................... ...................................................................................

................................................................................... ................................................................................... ................................................................................... ...................................................................................

................................................................................... ................................................................................... ................................................................................... ...................................................................................

................................................................................... ................................................................................... ................................................................................... ...................................................................................

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

666

6���@@I

���@@I

6

���6

@@I

ßuJ�à � G34 ¤ CK:<¥"LN?<8EAXL�=7�B4F=B:<4ID=�� G>CK¥7G34MI�=B:F4M6 i GNL�=7�B4 f � :�4M6 i G8-�>?<6�ID4 A+GB6�A i =>?325; J<4 H36�IæA=>kA+G34 ¤ CP:<¥�§ 8F894-�B4M:<�+L�=7�B4�H<6�A+GÉk5ID=BLJ<=�A�AD=BL 2K47kPA�AD=j?<H<H<4MIN2K47kPA i =BIE:<47I-¢� 4 H32P6 i 4¾A+G34 :B?<LFJ<47I =>k��a6�TB8A+G34 ¤ CK:<¥ i 6M:¿ID4M6 i G A+GB6�A�89�B?<6MID4 fß � 6 i G�:B?<LFJ<47IzLN6�T�J<4a=3J�AD67CK:<4M;NJMT6�;3;BCP:<¥�A+G34X:B?<L�J34MI�8�CP:aA+G34X89�B?<6MID4M8J<4�25=�� C�A�6�:<; GB6\�BCP:<¥�6M: 4M;<¥74Ü=BIi =BID:<4MIQCK: i =BLFLF=B:h��C�A+GÜC�A f à �S4�r:<;ÉA+GB67A"A+G347ID4�6�ID4127

�a6\T38�A+G34¤ CP:<¥ i 6M: ID4M6 i G A+G34 ?<H<H<4MI 254�kPAi =BID:<4MI�CP:N8-4-�B47:NL�=7�B4M8 f.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

K

1 1

2 2 1

4 5 3 1

9 12 9 4

21 30 25

51 76

127

�����

í f ß � =�AD43ì � =BI�A+G>CK8R�B?r478+ADCK=B:S6M:38E� 47I�8 i =B:>A�6MCP:BCK:<¥"4*ò�H<ID47898+C5=B:38O8*? i G�6M8 4π13

6�ID46 iEi 47HBA�6�J>254 f�� k�TB=>?�GB6��B4X6 i 6M2 i ?32K6�AE=BI TB=>?�LN6�T�?<8-4�A+G34XJB?BA�AD=B: ku=BIπCPk�TB=>? 2PC ¤ 4 f à

ßK6�à � G34��9=B:<478�k56�L�CP2 T�2@C��B478�CP:F6 H<4MI�ku4 i AD2 TF89�B?<6MID4 G3=>?<8-4>¢10L JMT

10L�¢>� G>C i GCK8�HB2K6 i 4M;�4*òM6 i AD2�TSCP:�A+G34FLNCK;<;>254�=>k�6

40L JMT

40L 25=�A-¢�4M:>ADCPID472 T i =7�B4MID4M;ßu4+ò i 47HBAaku=BIRA+G34SG3=>?<894\àXCK:�¥�I�6�8*8 fÉ� G34-T ¤ 4M4MH�A+G34Qk56�L�CP2 TwH<4�A-¢ Ù =>2P2 T�A+G348-G34M4MHy¢>AE4\A+G34MID4M;�AE=�A+G34 L�C5;3;B2K4�=>k�=B:<4 8+C5;34 =>k A+G34 G3=>?<8-4 =B:F6

15L ID=BHr4 f

�SGB67A�CK8$A+G34"6MID4M6�=>k�A+G34"H36�IæA�=>k�A+G34�2P6��a:QßKCP:�L 2 à�CP:N� G>C i G Ù =>2@2�TFCP8X6�J>254AE=�¥�I�67Þ*4 � ß � 4�4�8-GB6�;<4M;N6�ID476 f à

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

................................................................................................................................................................................................................................

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

6

?...........................................................

...........................................................

40m

� -..............................

.

..

.

.

..

.

.

..

.

..

.

.

..

.

..

.

.

..

.

.

..

.

40m

...............................................................................................................r٠=>2P2 T� G34��9=B:<4M8

6N

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

...............................................................................................................................................................................................................................

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

.. ....................................................................................................................................................r

Ù 67CK8�T� G34 � LNC�A+GB8

ßuJ�à � G34��9=B:<478�§y:<4�C5¥7G3J3=>?<I�8�¢0A+G34 � LNC�A+GB8�¢�GB6\�B4�6M:VCK;<47:>ADC i 6M2�2K=�A�AE=QA+G34��9=B:<4M8JB?BA�A+G34�CKIzG3=>?<8-4RCK8�2K= i 6�AE4M;��>�B4aLF4�AEID478zAE=�A+G34 � =BIæA+GN=>krA+G34 i 4M:>AEID4 f�� G347CPIHr4\A�89G34�47H�¢ Ù 67CK8�T ¢3CP8�AD4�A+G347ID4�;�AE="A+G34 LNCK;<;>254�=>k A+G34�89=>?BA+G34MID:�8+C5;34�=>kyA+G34G3=>?<8-4�=B:F620L ID=BHr4 f �SGB6�A�CP8�A+G34 6�ID476�=>k A+G34 H<6MI�A$=>k A+G34 2P6��a:�ßKCP:�L 2 àCK: � G>C i G Ù 6MCP8�T"CK8�6�JB2K4RAD=N¥�I�6�Þ*4 �

��� âæç�å é � å��3êKç � è fß56\à � G34FH36�IæAR=>k�A+G34N2P6��a:SCP:Q� G>C i G Ù =>2P2 Ti 6M:S¥�I�67Þ*4 i =B:38+CK8EAD8O=>k�6�894ML�C i CPI i 254"=>k�I�6�;>CP?<8

156�:<;FAæ� =Q�B?<6MI�AD4MI i CPI i 25478R=>k�I�6�;>CP?<810f � AD8O6MID4M6CK8zA+G347ID47ku=BID4 1

2π152 + 2 · 1

4π102 = 325

2πf

ßuJ�à � G34�H36�IæAO=>k�A+G34"2K6\�a:�CP:�� G>C i G Ù 67CK8�Ti 6M:�¥�I�6�Þ*4 CP8�LF6�;<4a?<H�k5ID=BL =B:<4�894ML�C i CPI i 254 6M:<;ku=>?<Iy=�A+G34MIy�>?<6�IæAE47I i CPI i 25478�¢\6�8�8-G3=��a: f�� AD806MID4M6�CP8A+G34MID4�ku=BID4 12π202+2· 1

4π152+2· 1

4π52 = 325π

f ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

...............................................................................................................................................................................................................................

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

.

.. ....................................................................................................................................................

.

..

.

..

.

..

..

.

..

.

..

..

.

..

...

..

..

..

..

..

..

..

...

...

...

...

...

...

.............................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.........................................................................

........................................................

...

..

..

.

..

.

.

..

.

.

.........................................................

................................................................................................................................

..

.

.

.

.

..

.

.

.

.

.

..

.

.

.

..

..

.

.

.

.

..

.

.

.

.

.

..

.

.

.

..

r

Ù 67CK8�T� G34 � LNC�A+GB8

6N

� GB6�A�J3I�CK:<¥�8�?<8zAE= A+G34 47:<;F=>ky6�:<=�A+G347I�CP898+?r4 f 132K4M6M8-4R8-47:<;�CP:N8-=>2@?BADC5=B:38��

�����

� ������� � ��� ��¿�� � ������ ��w67A+G347LF6�ADC i 6M2��w6\T<G347L¶J34�¥�6M:FCK: ����� � 6�8$ô�lhô_õ �Bï� �ô_õ � �9ô7~��E}Mð �+îBô7~Bóæ} ��ô7î3ñ�� �p ��� ��� ���B}B}�~<ô7î3ñ��0î � �>ï �+ö � õ ���-õ+ð<ñBï�îMõ*ö f � A i =B:>ADCK:B?r478�¢���C�A+G"A+G34a896ML�4 47LFH<GB6�8+CK8\¢6�8�6�:�CP:>AE4M¥�I�6M2rH<6MI�A�=>k���^������ �"!$#rZ\[��"!&%('_WM^)�7[+*,%-!.#�� �"!.#rZ_[��"!(%&'/��01� �"23#rZ\[ f� G34Q�w6\T<G347L � ;BC�AE=BI�CK8 �94 � � =3=BH<4MI�ß g�i 6�;>CK6 Ú :BC��B47I�8*C�A�T�à f�� G34 g 898+CK8EAD6M:>A�w6�T3G34ML � ;>C@AD=BI�CP8 � 6�:a¦�6M:<;<47I_�<?<ID¥7G"ß Ú :BC��B47I�8*C�A�T"=>k ��67AD4MI�25=3=�à f0� G34R=�A+G34MI�8EAD6 �L�47L�J34MI�8�6�ID4 �9=3GB:54�I�6M:>A�� i ¡M=>?r¥7G>2PCP: ß Ú :BC��B47I�8*C�A�T�=>k � 4\���rID?<:38D��C i*¤ à+¢>��=B:BC ¤ 6��G3J347CP8�ß � 6�A+G34MI���C i GB6�4�264�=34\A�Þ � 4 i =B:<;36MI�T ��i G3=3=>2�¢z��CP898+CK8*8967?r¥�6�àE¢ � I�C i87 =3J<47I�Aß�¡M4M= � 6�TB478 � C5¥7G ��i G3=3=>2�¢ � ID4M;<47IDC i AE=B:�àE¢�¡76�IDI�T 7 C i 4�ß Ú :BC��B47I�8*C�A�T�=>k ��6�AE47ID2K=3=�à+¢6�:<; 7 =B:S¡76�: i 6�8EAE47I�ß Ú :BC��B47I�8*C�A�TN=>k � =BID=B:>AD=�à f

9 :8;=<?>�@ ACBEDGFIHJ>�@LK

MON�PRQ S S-NUT5VOWXP)Y[Z]\&^�V3YU_`N�Z(Z]\aNcb)W)Y`YRWdS&P3ZeQ-W�VJY`^XPgf�hi\aWJj)S�kNR_`N/Y8lOP[hi\XmNgYRN/ViZEV�P)_?mN/\aW^Jbg^�VOZ�S-Non3paqRrts qRpuRqRn)vwqdrtx�paq[ydz�z3{X|�}3N/YtY/WdS&PJZeQ-W�V3Y~\aNJ�� P3N/Yt^Rhi\OkN/Y � N�Z(Z-N�l�^JZ&N5ViN�Y/NR\aW�VOZhi\(Q�Y/N/Y�N/V � W�_h�Z&NE�OP3N6Y)� Q SOViWXPdY�\aN/Y/Z&NElOPoZ-NR_hOY�^JbU^�ViZiS�^�h�PJj)S Q � ^JZeQ-W�V5l"NgY�YRWdS&P3ZeQ-W�VJYd|�X� ^3�OPJN�hi\aWJj)S)kN/_�NcYRN/\&^Ih�PJj)S QRmNIl�^�VJY�S-N/Y�l"N�P/fIS�^�V3�XP3N/Y�W)� � Q-NUS S-N/Y�lOP � ^�VJ^3l�^� ^�V3�dS�^dQ�Y~N�Z��e\&^�V��� ^dQ�Y&�/|1��^�V3Y�S-NgYV�P)_?mN/\aW)Y~�g���R���8N�Z,�w��S$� ^�V��dS�^dQ�YEhi\XmN � mN)l"N/\&^�S-Nt��\&^�V��� ^dQ�Y.�N�Z�l�^�V3Y�S-NgY�ViP�_?mNR\aW)Y��R�i�)����N�Z� /�OS-N���\&^�V��� ^dQ�Y1hi\XmN � mNdl"NR\&^,S$� ^�V3�dS�^dQ�Yd|})^c\XmN)l�^ � ZeQ-W�V�Y/WXP � ^dQ-Z&N8\aNR_`NR\ � Q-NR\�¡dN/^�VO¢w£�^�\ ��¤ N/\.\&Q-NR\�N�Z£�^�\wZeQeV¦¥�WdS]l�YRZ-NgQeV���l"NS$��§¨V�Q-b)N/\&YaQ-Z)mN,l"N�£IW�VOZ]\XmNg^dS ��l�� ^Jb)WdQe\�Z]\&^3lOPRQ-ZOS-N/Y1hi\aWJj)S)kN/_�NgY)|

µ ¯© ë f�ª ä � � � é�«ã ���ät¬ ä �>â+ã � � O­ ��ã�ä �®0è>ç$¯Mã\ä�é+çæê3«ã° ã�± � ä�ç�å �<ã[²7ã�äEä+ã"³g´aã �O¯Mã � � êwµX¶ � �<è�· é ��´`¸ f£ :Ü;34M8*8*CP:<4w:<47?3k i 4MI i 2K4M8 i =B:>ADC5¥\?<8S;34�I�6\TB=B:

1/24\A�A�6�:<¥747:>AD8 Ý6�?<: i 4MI i 2K4 ;34aI�6\TB=B:

1f� ID=>?>�B4MIn2K6°;>CK8EAD6M: i 4°47:>AEID4 25478 i 47:>AEID4M8';>?H<ID47LNCK4MI�4\A�;B?�;34MID:BC547I�;<4 i 478 i 4MI i 2K4M8 f

.

..

..

.

..

..

..

..

..

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...........................................................................................................................................

...

...

...

...

...

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

...

...

...

...

...................................................................................................................................................................................................... .

..

..

.

..

.

..

.

..

..

..

..

..

..

..

..

.......................................................................

...

...

..

..

.

..

..

..

..

.

...

...

....................................................................................................

..

..

..

..

..

.

..

.

..

..

..

..

..

..

..

.......................................................................

...

...

..

..

..

..

.

..

..

..

...

...

...................................................................................................

.

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

.......................................................................

...

..

..

..

..

..

.

..

..

...

...

....................................................................................................

.

..

..

.

..

.

..

.

..

..

..

..

..

..

..

..

............................

.........................................................................................................................

................................................

..

..

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

.....................................................................

...

...

..

.

..

..

..

..

.

...

...

.....................................................................................................

..

.

..

..

..

.

..

.

..

..

..

..

..

..

..

..

.......................................................................

...

...

..

..

.

..

..

..

..

..

...

.....................................................................................................

.

..

..

..

..

..

.

..

..

..

..

..

..

..

..

.........................................................................

...

..

..

..

..

..

.

..

..

...

...

....................................................................................................

..

..

.

..

..

.

..

.

..

..

..

..

..

..

..

..

.......................................................................

...

...

..

..

.

..

..

..

..

..

...

.....................................................................................................

..

.

..

..

..

.

..

.

..

..

..

..

..

..

..

..

..

.....................................................................

...

...

..

..

.

..

..

..

..

..

...

.....................................................................................................

....................................................................................

µ ¯© í f�ª ä � � � é�«ã ���ä6¬ ä �>â+ã � � O­ ��ã�ä ��®0è>ç$¯Mã\ä�é+çæê3«ã ° ã�± � ä�ç 7å �<ã5²7ã�äEä+ã"³g´aã �O¯Mã �� êwµX¶ � �3è�· é ��´`¸ fÚ : i 6�IDI��4

ABCD4M8EAQCK:38 i IDC�AS;36M:38V?<:'894 i �AE4�?<I i CKI i ?32P6MCPID4 i =>?>�BI�6�:>A�254�G>?3C�ADCDÝ47L�4�;y§ ?<:n;>CK89�B?r4;<4�I�6�TB=B:

1f�Ù 47?�ò�;34M8R89=BLFLF4�A�8R;B? i 6�IDI��4�89=B:>AX8+?<I25478�I�6�TB=B:38a;>?�J3=BID;�4\AX2K4M8�;<4�?�òQ67?BAEID4M8�89=B:>AX8*?<I�2K4i 47I i 254 f� ID=>?>�B4MIr2�§ 67CKID4�;B? i 6MIEI��4 f�£ :R4*òMC5¥74�?<:<4�I��47Hr=B:3894

4*òM6 i AE4�;<4z2P6�ku=BIELF4 a + b√

c

d

¢�=�Ý?a¢b¢c4�A

d8-=B:>Ay;<4784M:>ADCK4MI�8 f .

..

..

.

..

..

..

.

..

..

.

..

.

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

.

..

..

..

.

..

..

..

.

..

..

..

.

..

..

.

..

..

..

..

..

..

.

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

��� �

µ ¯©O² f¨ª ä � � � é�«ã ��Mä�� �è>ç�ã-å�²_é 7ç �t«ã�ê � ��ç �èyê��² �ç ��ã-ç��,±�ã�ä�ç�â�Mè � â � � � å ��² 7ç ��ã_ç �² �ç$­ Mè f

1B=>?<Ik = 1

¢2¢

. . .¢

n¢z8-=>C�A�2K4M8NHr=>CP:>AD8

ak = (k, n)4\A

bk = (k, 0)4�AR8+?<H<H<=B8-=B:38 �B?r4 i GB6��B?r4 i =>?<HB254ak¢bk4M8EARID472@CD�4FH<6MIO?<:�894�¥�LF4M:>A�;34Q;3ID=>C�AE4�B4MIæADC i 6M2 f � ?<H<H<=B8-=B:38��B?�§ =B:N;34M8*8*CP:<4X?<:�:<=BLFJ3ID4R6MI+J>C@AEI�67CKID4X�r:BC<;34R894�¥�LF4M:>A�8�G3=3�IDC�Þ*=B:>AD67?�òVID472@CK6M:>A ;<4�?�ò��B47I�ADC i 6725478"6�;�÷u6 i 47:>AE478�;<4�89=BI�AD4S�B?�§ 6M? i ?<:�Hr=>CP:>A�; § ?<:<4�B4MIæADC i 6M2K4 �>?r472 i =B:<�>?r4a8-=>C�A�?<:<4 4+òMAEI��4ML�C@A��4a;34 ;347?�ò�894�¥�LF4M:>A�8�G3=BI�C@Þ*=B:>A�6M?�ò f£ :S;��4��r:BC�A�?<:<4�6MH<HB2PC i 6�ADC5=B:S;34"2�§ 47:38-47L�J>254

A = {a1¢a2¢. . .¢an} ;B6�:382�§ 4M:3894MLFJB2K4

B = {b1¢b2¢. . .¢bn} ;34a2P6 LF6M:BCDÝ47ID4 8+?3C��>6M:>AE4Nì>H36�IæAD6M:>A$;34 ai

¢<=B:;<478 i 4M:<;�47:�8*?3C �>6�:>AX254F894�¥�LF4M:>A�÷5?<89�B?�§uÝ6 i 4��>?�§ =B:�ID4M: i =B:>AEID4F2�§ 4*ò7AEI��47LNC�A7�4�;y§ ?<:8-4M¥�L�47:>A�G3=BI�C@Þ*=B:>A�6M2�¢��B?�§ =B:"8*?3C�A�6M2K=BI�8�÷5?<89�B?�§uÝ6�89=B:�6M?BAEID4R4+òMAEI��4ML�C@A��4>¢�=B:�;34M8 i 47:<;4M:38+?3C@AD4F2K4N25=B:<¥F;>?�:<=>?>�B4M67?w894�¥�LF4M:>A$�B4MIæADC i 6M2�4\AR6MCP:38*C�;<4F8*?3C�AE4>¢0H<=>?<IR6�IDIDC �B4MI�r:3672547L�47:>A�Ý6bjHr=>?<I�?<: i 47I�A�6MCP:

jf ��=B:>AEID4MIO�>?r4F;<4�?�ò�H<=>CK:>A�8a; §

A:y§ 6MIEI�C��B47:>AH<6M8$67?FL Û47L�4�Hr=>CP:>A�;<4

Bf

µ ¯©�� f[ª ä � � � é�«ã ��Mä�¶ � �<è���ä Mè�ê ° âR¸ � �� �Bå@ç5è �¦®0è>ç$¯Mã\ä�é+çæê3«ã � � ´ � �i¯Mã �i³¬ ä ��èBé�­�ç�â� � ��ä+ã��<ã\ä�ç�â9ê � è���´`¬ f

¡M4M8�:<=BL�JBID4M834543

4�A713317

89=B:>A$;<478X:<=BLFJ3ID478XH<672PCP:<;3ID=BLF4M8�H3?3CP8-�>?r4254�?<I�8 i G>C ��ID478$6�H3H<6MI�6MCP89894M:>A�;36M:38$2K4aL Û47L�4 =BID;3ID4>¢<�B?r4�2��>?r4 89=>C@A�2K4a8-47:38�;<4�254 i �AD?<ID4 f ��=B:>AEID4MI��B?r4�AD=>?<8�2K4M8�:<=BLFJ3ID478zH<672PCPL�;BID=BL�478�;34R�B?<6�AEID4 i G>C��yID4M8�8-=B:>A�;34M8LN?32�ADCKHB25478�;3411f

µ ¯©�� f[ª ä � � � é�«ã ��Mä�¶ � �<è���ä Mè�ê ° âR¸ � �� �Bå@ç5è �¦®0è>ç$¯Mã\ä�é+çæê3«ã � � ´ � �i¯Mã �i³¬ ä ��èBé�­�ç�â� � ��ä+ã��<ã\ä�ç�â9ê � è���´`¬ fßK6�à � A�6�:>A�;<=B:3:��4V?<:j:<=BLFJ3ID4�H<672PCP:<;3ID=BLF4w;34��B?<6�AEID4 i G>C ��ID478�¢��B?r4�2P2K4�4M8EA�2K6H<ID=3JB6�J>CP2@C@A��4a�>?�§ C@2 8-=>C�A�?<:FL�?32@ADCPH32K4 ;34

99 �ßuJ�à � A�6�:>A�;3=B:3:��4F?<:�LN?32�ADCKHB254�;<499¢��>?r472@254�4M8EAO2P6�H<ID=3JB6�J>CP2@C@A��4��B?�§ CP2�8-=>C�AO?<::<=BL�JBID4aH36M2@CK:<;BID=BL�4 �

µ ¯©o© f1ª ä � � � é�«ã ���ä~²\çæê ���O¯ � è��ä � ��� � ± «Mèyã��é�êKç ��� � �J±��è>ç�ã f� =>C@A

a¢b4\A

cAEID=>CP8R:<=BL�JBID4M8RI��4�4�2K8OHr=B8+C@ADC@k58XAD472P8��>?r4

ab + bc + ca = 3f��=B:>AEID4MI��>?r4

ab

c2 + 1+

bc

a2 + 1+

ca

b2 + 1≥ 3

2

fµ á�±O± f1ª ä � � � é�«ã ���ä���ã ��� ä+ã ���oµ���Mè � �åPå ��� J± �<ã�è ���1² � «� ³3® f

Ù 6�:38a?<:VAEI�CK6M:<¥\254Q�B?r4�2 i =B:<�B?r4ABC

¢�89=>C@AD?<:�H<=>CK:>A�8+?<IO254 i7Û=�A��4

AC¢

E?<: H<=>CK:>Aw8+?<I

AB¢"4\Aw8-=>C�A

P2K4�H<=>CK:>A�; § CP:>AE47I�8-4 i ADC5=B: ;34

BD4\A

CEf

� CAE : EB = r

4\AAD : DC = s

¢OAEID=>?>�B4MIS254nI�6�H3Hr=BIæA�;34M8�6MCPID4M8[ABC] : [PBC]

47:�ku=B: i ADC5=B:F;34r4�A�;34

sf

f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

�����

µ ¯© ë fEª ä � � � é�ã � ����¬ ä �BâEã � � O­ ��ã�ä � ° ã�± � ä�ç�å®0è>ç$¯Mã\ä�é+çæê � ��� ´aãJ­ ��� �Mè ��å �è�� � � êwµX¶ � �3è�· é ��´`¸ f

� CP:<4 i CKI i 2K4M8�=>k<I�6�;BC@?<81/2

6MID4O4*ò7AE47IE:3672P2 T�AD6M:<�¥74M:>A�AE=�6 i CKI i 2K4 =>kyI�6�;BC@?<816M:<;N6�ID4RA�6�:<¥747:>A�AD=N=B:<46�:<=�A+G347I-¢B6�8�89G3=��a: fÙ 4�AD4MIDLNCP:<4�A+G34O;BCP8+A�6�: i 4RJ34�Aæ� 4M4M:aA+G34 i 47:>AEID4M8=>krA+G34��<I�8+Az6M:<;�2P6�8EA�=>krA+G34 i CKI i 2K4M8$=>k I�6�;>CP?<8

1/2f

.

..

..

..

.

..

..

..

..

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...................................................

..............................................................................................................................................................................................................................................

...................................................................................................... .

..

..

..

..

..

.

..

..

..

..

..

..

..

..

.........................................................................

...

..

..

..

..

..

..

.

..

...

...

....................................................................................................

.

..

.

..

..

..

.

..

.

..

..

..

..

..

..

..

.......................................................................

...

...

..

..

..

.

..

..

..

..

...

...

...................................................................................................

..

.

..

..

..

.

..

.

..

..

..

..

..

..

..

..

........................................................................

...

..

..

..

..

.

..

..

..

..

...

.....................................................................................................

.

..

..

..

..

..

.

..

..

..

..

..

..

..

..

.........................................................................

...

..

..

..

..

..

..

.

..

...

...

....................................................................................................

..

..

..

..

..

.

..

.

..

..

..

..

..

..

..

........................................................................

..

...

..

..

..

..

..

.

..

..

...

...

...................................................................................................

..

..

..

.

..

.

..

.

..

..

..

..

..

..

..

..

.......................................................................

...

...

..

.

..

..

..

..

.

...

...

.....................................................................................................

.

..

..

..

..

..

.

..

..

..

..

..

..

..

..

.........................................................................

...

..

..

..

..

.

..

..

..

..

...

.....................................................................................................

..

..

..

.

..

.

..

.

..

..

..

..

..

..

..

..

.......................................................................

...

...

..

.

..

..

..

..

.

...

...

.....................................................................................................

..

..

.

..

..

.

..

.

..

..

..

..

..

..

..

..

..

.....................................................................

...

...

..

.

..

..

..

..

.

...

...

.....................................................................................................

...

.................................................................................

µ ¯© í f8ª ä � � � é\ã�� ����¬ ä �BâEã � � O­ ��ã�ä � ° ã�± � ä�ç 7åo®�èBç(¯Mã�äæé*ç�ê� ��� ´aãJ­ ��� �Mè ��³åMè � � � êwµX¶ � �3è�· é ��´`¸ f� �B?<6MID4

ABCDCK8zCP:38 i I�C5J34�;"CP:N=B:<4��*47CK¥7G�A+GN=>k6 i CPI i 254�=>k�I�6�;>CP?<8

189=FA+GB6�A$A+G347ID4"CP8R=B:<4 �B47I�AD4*ò�=B:4M6 i GNI�6�;BC@?<8$6M:<;"Aæ� =a�B4MIæADC i 4M8$=B:"A+G34a6MI i�fÙ 4�AD4MIDLNCP:<4�A+G34�4*òM6 i A 6�ID476$=>k7A+G34�89�B?<6MID4zCK:�A+G34

ku=BIEL a + b√

c

d

¢�� G34MID4a¢b¢c¢B6M:<;

d6�ID4RCK:>AD4�¥747I�8 f

.

..

..

.

..

..

..

.

..

..

.

..

..

.

..

..

.

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

.

..

..

..

.

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...

....

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

µ ¯©O² f,ª ä � � � é\ã�� � � � �è>ç�ã-å�²_é 7ç �ré�ê � �<ã\è�ê��² 7ç ��ã_ç��,±�ã�ä�ç�â�Mè � â � � � å ��² 7ç ��ã_ç �² �ç$­ Mè f

¡M4�AnJ34�6�H<=B8*C�ADC��B4OCP:>AE4M¥74MI f�� :�A+G34a� 6�IæAE478*CP6�:�HB2K6M:<4B¢ i =B:38*CK;<47I�A+G34RHr=>CP:>AD8

ak = (k, n)6M:<;

bk = (k, 0)ku=BI

k = 1¢2¢

. . .¢nf �S4 i =B:3:<4 i Aa4M6 i G�H<67CKI

ak¢bkJMT�6a8EAEI�6MCK¥7G�A�ß��B4MIæADC i 6M2 à02PCP:<4�8-4M¥�L�47:>A f�� G347:�� 4a;BI�6��'6�:�6�IEJBC�AEI�6�I�T"�r:BC�AE4:B?<L�J34MIa=>k�G3=BIDC�Þ*=B:>AD672�2PCP:<4Q894�¥�LF4M:>A�8�¢�476 i G i =B:3:<4 i ADCP:<¥NAæ� =V6�;�÷u6 i 4M:>AX�B4MIæADC i 6M22PCP:<4�8-4M¥�L�47:>AD8\¢�8*? i G�A+GB6�A�:<=�=B:<4�H<=>CK:>A�=B:�6�:�TQ�B4MIæADC i 6M2�8-4M¥�L�47:>ARCP8�A+G34�4M:<;3�Hr=>CP:>A�=>krAæ� =NG3=BI�C@Þ*=B:>A�6M2r894�¥�LF4M:>A�8 f¡M4�A

A = {a1¢a2¢. . .¢an} 6�:<; B = {b1

¢b2¢. . .¢bn} fFÙ 4��r:<4"6�LF6MHk5ID=BL

AAE=

B6�8"ku=>2@25=��a8�ìF8+A�6�IæADCK:<¥Qk5ID=BL

ai¢zAEI�6\�B472$;3=��a:wA+G34V8-4M¥�L�47:>A ?<:>ADCP2TB=>?SLF4�4\A�A+G34�47:<;<�+Hr=>CP:>AX=>k�6NG3=BI�C@Þ*=B:>A�6M2�894�¥�LF4M:>A-¢y¥7=NAE=NA+G34�=�A+G34MI$4M:<;3�*H<=>CK:>A=>kyA+GB6�A�894�¥�LF4M:>A-¢<6M:<; i =B:>ADCK:B?r4�=B:�;<=��a:�A+G34 :<4\�'�B47I�ADC i 672 2@CK:<4>¢<ID47Hr4767ADCP:<¥RA+G>CK8?<:>ADCP23A+G34MID4�6MID4a:<="L�=BID4aG3=BI�C@Þ*=B:>A�6M2<8-4M¥�L�47:>AD8zAD=�LF4�4\A-¢>�<:36M2@2�T�4M:<;>CK:<¥R67A

bjku=BI8-=BLF4

jf � G3=���A+GB6�Az:<=�Aæ� =�H<=>CK:>A�8�=>k

ALN6�H"AE= A+G34a896ML�4�H<=>CK:>A�=>k

Bf

µ ¯©�� f6ª ä � � � é\ã�� ����¶ � �<è ��ä Mè�ê ° âR¸ � � ���>åPçKè���®0è>ç$¯Mã\ä�é+çæê � ��� ´aãJ­=¬ ä ��èBé�­�ç�â� ���ä+ã��<ã\ä�ç�â9ê � è���´`¬ f

� ?<LFJ<47I�8z8+? i G 6M834543

6�:<;713317

� G3=B894O;>C5¥\C�AD8 i 6M:"J34XID4-�B4MI�894�;R��C@A+G3�=>?BA i GB6�:<¥\CP:<¥NA+G34Q:B?<LFJ<47Ia6MID4 i 6M2@254M; ��7å@ç5è��>ä � ±�ã_é f � G3=��¿A+GB67A�672P2�ku=>?<IE�9;>C5¥\C�AH<672PCP:<;3ID=BLF4M8�6�ID4�LN?32�ADCKHB25478�=>k11f

µ ¯©�� f6ª ä � � � é\ã�� ����¶ � �<è ��ä Mè�ê ° âR¸ � � ���>åPçKè���®0è>ç$¯Mã\ä�é+çæê � ��� ´aãJ­=¬ ä ��èBé�­�ç�â� ���ä+ã��<ã\ä�ç�â9ê � è���´`¬ fßK6�à[4�C �B4M: A+GB6�A�6R:B?<L�J34MI�CK8z6Oku=>?<I+�*;BCK¥\C@A�H<672PCP:<;3ID=BLF4B¢7� GB67A�CK8�A+G34RH3ID=3J36�JBC@2PC�A�TA+GB67A�A+G34�:B?<L�J34MI�CP8�6 L�?32@ADCPH32K4 =>k

99 �ßuJ�à�4�C �B4M:FA+GB6�A�6�ku=>?<I+�*;BCK¥\C@A$:B?<LFJ<47I$CP8O6"LN?32�ADCKHB254"=>k99¢<� GB6�A�CK8$A+G34"H3ID=3J36��JBC@2PC�A�T�A+GB6�A�A+G34a:B?<LFJ<47IzCP8�6aH<672PCP:<;3ID=BLF4 �

��� �

µ ¯©o© fEª ä � � � é�ã � ����²\çæê ���O¯ � è��ä � ��� � ± «Mèyã��é�êKç ��� � ±�MèBç f¡M4�A

a¢b¢B6�:<;

cJ34�Hr=B8+C@ADC �B4�ID476M2r:B?<LFJ<47I�8z��C@A+G

ab + bc + ca = 3f 1<ID=7�B4A+GB67A

ab

c2 + 1+

bc

a2 + 1+

ca

b2 + 1≥ 3

2

fµ á�±O± fEª ä � � � é�ã � ��� ��ã ��� ä+ã ���oµ���Mè � �åPå ��� J± �<ã�è ���1² ��® � � f

¡M4�AABC

J34�6�:N6MI+J>C@AEI�6MI�T"AEI�CK6M:<¥\254 f ¡M4�AD6�:<;

EJ34aH<=>CK:>A�8$=B:"A+G34�8*CK;<478

AC6�:<;

AB¢$ID4M8*Hr4 i ADC��B4�2�T ¢$6M:<;É2K4�A

PJ34�A+G34�H<=>CK:>A�=>k$CK:>AD4MI�894 i ADCK=B:j=>k

BD6�:<;CE

f � kAE : EB = r

6M:<;AD : DC = s

¢ ;34�AD4MIDLNCP:<4aA+G34 I�67ADCK=�=>k�6�ID476�8[ABC] : [PBC]

CP:"AE47IELN8�=>kr6M:<;

sf

9 :�;¦<?>�@ �[DIH������3D��?K

µ ¯ ëOë fEª ä � � � é�ã � ��� ° � � J±�±�ã���� Mé*é*ç@å � � êuä�Mé�� � ��ä ��� ��ä MèyâEãOµ¡M4�A

ABCDJ<4a6 i =B:��B4*ò��>?<6�;BIDC@2K6�AE47I�6M2�¢<6M:<;N254\A

P¢Q¢R¢SJ<4RA+G34 LNCK;<�Hr=>CP:>AD8O=>k

AB¢BC

¢CD

¢DA

¢rID4789H<4 i ADC �B472 T f � ?<H<H<=B8-4aA+GB67A�ku=>?<I$;>CK8EADCK: i A�2@CK:<4784M6 i G�H36�8*8*CP:<¥NA+GBID=>?r¥7Gw=B:<4Q=>kP¢

Q¢

R¢Si =B: i ?<I�6�A�6�Hr=>CP:>A

OfnÙ I�6\� 2@CK:<478H<6MI�6M2@254�2�AE=$A+G34M894�ku=>?<Ir2PCP:<4M80JB?BAyH36�8*8*CP:<¥�A+GBID=>?r¥7GXA+G34�LNCK;<�+Hr=>CP:>AD8�=>k7A+G34$=BH3Hr=B8+C@AD48*CK;<478 f 1<ID=7�B4RA+GB6�A�A+G34M894Rku=>?<I�2PCP:<4M8�6MID4a672K89= i =B: i ?<IDID4M:>A f

� â � ± �MçKè BêKç � è ��� é � å �BêKç � èBé � � � Mé �è �Xã�è 3ã\ä �ué�êMè � ��å �[² ��ä Bã ����Mè � ²�ç�ê ��O¯ � è��ä � ��� � ± «Mèyã��é�êKç � � � ±��è>ç f

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................

...........................

............................

............................

...........................

............................

............................

...........................

............................

............................

....................................................................................................................................................................................................................

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

............

..

..

..

..

.

..

.

..

..

..

..

..

..

..

..

.

.

.

..

..

.

..

..

..

..

..

..

..

.

..

.

..

..

..

.

..

..

.

..

..

..

..

..

..

..

.

..

.

..

..

..

..

..

..

..

..

.

.

.

..

..

.

..

..

..

..

..

..

..

.

..

.

..

..

..

.

..

..

.

..

..

.

.

..

..

..

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...

..

.

.

..

..

.

.

..

..

.

.

..

..

.

.

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

.

.

..

..

.

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

..

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................A

B

CD

M

O

O1, O2

P

Q

R

S

� G34NH<=>CK:>A�8P¢Q¢R¢Sku=BIDL 6FH36�I�672P2K472K=3¥�I�6�L�¢08+CK: i 4"A+G34-TS6MID4"A+G34NLNCK;<�Hr=>CP:>AD8�=>kzA+G34�8*CK;<478�=>k�6V�B?<6�;3I�CP2P67AD4MI�672 fÉ� G34MID4�ku=BID4B¢�A+G34V;>CK6�¥7=B:36M2P8�=>k

PQRSJBCP8-4 i A�4M6 i GN=�A+G34MI�67A�6 H<=>CK:>AMf¡M4�A

O1J34�A+G34QHr=>CP:>A =>k�CP:>AE47I�8-4 i ADC5=B:w=>kzA+G34QH<6MI�6M2@254�2�AE=

ORA+GBID=>?r¥7G

P6�:<;FA+G34"H<6MI�6M2@254�2yAD=OP

A+GBID=>?r¥7GRf � CP: i 4

ORCK8XH36�I�672P2K472yAD=

PO16�:<;

OPCP8

��� �

H<6MI�6M2@254�2BAD=RO1

¢>�B?<6�;3I�CP2P67AD4MI�672ORO1P

CP8�6RH<6MI�6M2@254�25=3¥�I�6ML f � CKL�CP2P6�I�2�T ¢�CPkO2CP8A+G34 H<=>CK:>A�=>k CK:>AD4MI�894 i ADCK=B:F=>k A+G34aH36�I�672P2K4723AE=

OSA+GBID=>?r¥7G

Q6�:<;"A+G34aH<6MI�6M2@254�2<AD=

OQA+GBID=>?r¥7G

S¢�A+G34M:

OSO2QCK8�6aH<6MI�6M2@254�25=3¥�I�6ML f� =���¢�A+G34Q;BCP6�¥7=B:3672K8�=>k�H<6MI�6M2@254�25=3¥�I�6ML

ORO1PJBCP8-4 i A�6�A�Hr=>CP:>A

M6M:<;

|OM | = |O1M | ¢<� G34MID476�8XA+G34N;>CK6�¥7=B:36M2P8R=>k�H36�I�672P2K472K=3¥�I�6�L OSO2QJ>CK894 i A�6�AHr=>CP:>A

MID478*?32�ADCK:<¥OCP: |OM | = |O2M | f��=B:3894��>?r4M:>AD2 Ty¢

O1 = O2¢B6M:<;"A+G34�ku=>?<I�2PCP:<4M8�6MID4 i =B: i ?<IEID47:>A f

��� �������������� ����������������������! #"�"�$%�'&�(')+*'�-,#&.����0/'�12�3���$4����$657"��98;:9 �<>=3=#�� �?�@��AB��C!�9�ED#D.$@�@&�FG�3�(#$H�7I2$J5K"L��8���C!�M�� ��4NO59C��E��<9$E��PRQ#�3�(�P $!D.S)���T"B)+SU�('P S V')��%W2CX�MI.12��(KY�$"#)+*#SU��$9�RP &�� �.8'&�('�Z�K�EC!�+ [D\I�"L��� $'��P�QB�3�('P $>IL&�S ]KLSB�>FG�1RSU)+&�(\"B)R*��#��� $>IL&�S ]KLS $�IL&�S ^E&�(7�µ ¯ ë í fEª ä � � � é�ã � ��� �� � � ç@åPå ��ä � ¯Mã �Z_3çPé7ê ���������® � �oµ

4�C �B4M:�CP8-=B8 i 4�25478�AEI�CK6M:<¥\254ABC

��C@A+GAB = AC

¢>2K4�A�A+G34�Hr=>CP:>AD8D6M:<;

EAEIDCP8-4 i A$A+G34�A+G>CKID;�8*CK;<4BC ` A+GB67A$CK8\¢ BD = DE = EC

f � =BI�8*LF672P206�:<¥\2K4M8A¢C@Az6MH<H<4M6MI�8$6M8�CPk

∠ACP8zAEIDCP8-4 i AE4M;NJMT"A+G34a894�¥�LF4M:>A�8

AD6�:<;

AEf 1<ID=M�B4OA+GB67A-¢�AD=A+G34 i =B:>AEI�6MI�Ty¢

∠BACCP8�:<4-�B4MI�AEI�CK894 i AD4�;NJMT"A+G34�8-4M¥�L�47:>AD8

AD6M:<;

AEf

��� å��3êKç � è ��� � 7é Mè �Xã�è 3ã\ä � 5é7ê �è � �7å ��² ��ä Bã � f¡M4�A

AB = AC = b6M:<;�2K4�A

AD = AE = yf �r25476�I�2�T ¢ 4ABD

CK8i =B:<¥�I�?r4M:>A�AD= 4ACEf ¡M4�A

∠BAD = ∠EAC = α6M:<;�2K4�A

∠DAE = βfz� =��AEIDCP6�:<¥\2K4M8

ABD¢ADE

6�:<;ACE

6M2@2>GB6\�B4�A+G34$8*6�LF4$6�ID4763¢78*CP: i 4�A+G34-Ta4M6 i GaGB6\�B4A+G34a896ML�4�672@ADC�AD?r;<4�6�:<;NJ36M8-4 f�� G347ID47ku=BID4>¢12by sin α = 1

2y2 sin β

¢� G>C i GV8*CPLFHB2PC@� 478RAE=

sin α = (y/b) sin βf � CK: i 4

y/b < 1¢�� 4 i =B: i 2@?r;<4NA+GB67A

α < βf��� ���a�����U����O ��O���R���E�E���b���J�H B"�".$G�'&�(')+*'�c,#&��U���d/'�12�3���$e����$f57"��98g:9 [<�=#=3�� [?O�M�

AB��C!�9�ED#D.$7�M&�FG�3�(B$9�'Ih$�5K"L��8��EC!�M�� .��NO59C��E��<9$[��PRQ#�3�(�P $7D.S)���i"#)S�(�P S V7)��jW2CX�'I�12��(9Y�$�"#)+*#SU��$�RP &�� �.8H�K�EC!�+ [D�I�"��E� $���PRQ#�3�(�P $HI�&.S ]KLSK��FG�1RS)&�(f"B)R*��#��� $XIL&�S ]KLS $HIL&�S ^E&�(�8@&�(��dI��2I�5lkB/'<�C�����5�$����F;m&�(�3n��P S $K�9��FZ&�(BS &

(P+^@�!�����+QBP SU��(3�

)�

o �#PU*GA'&�('��&.� ��&�(��Hk�����(#&�1LQH�2*��B^@��\P *3&BP�P *'!PU1RS &�(Bp��U@('���H('�#P� 3MS ������)�L�U��MP+�HFZ&�q3!PU*�LS 1&�1+p#Q�FG�(�P ��^@��1�q��µ ¯ ë ² fEª ä � � � é�ã � ���Nê� ã ° � �yã�± � ê � µ

� 47:NH<=>CK:>A�8�6�ID4R6�IDI�6�:<¥74M;�CP:�6�H32P6�:<4R8-=aA+GB67A�:<= A+GBID4M4�6MID4 i =>2P2@CK:<476�I f �SGB6�ACK8OA+G34NLN6\òMCKL�?<L :B?<L�J34MIR=>k�8-4M¥�L�47:>AD8OA+GB6�A i 6�:�J34F;3I�6\�a:Q÷�=>CK:BCP:<¥�Aæ� =S=>k�A+G34Hr=>CP:>AD8�8*? i G�A+GB6�Az:<=�A+GBID4�4 =>k<A+G34M894aH<=>CK:>A�8$6MID4a672P2B÷�=>CP:<4�;�AE="ku=BIDL¿6RAEI�CK6M:<¥\254 ���� å��3êKç � è ��� � ��ä�ç�ã_å���ä�ç]± 3ã\ä �Ré7ê � �<ã�èyê� �è�� � �Mé�êO¯ � ��ä�ç]± 3ã\ä ��®0è>ç$¯Mã\ä�é+ç�� ���� � � � ��¬ �Bã\è � é ��çKä*ã_é � ��ä �Bã\è�êKçKè f

�S4�?<894�A+G34zAD4MIDLNCP:<=>25=3¥-TR=>k>¥�I�6�H<GXA+G34M=BI�T f ¡M4\AGJ<4�6�¥�I�6MHrGR=B:

n�B4MIæADC i 4M8 f� G34Q894�A�8

V6�:<;

EID47H<ID478-47:>ARA+G34�894�A�=>k��B47I�ADC i 478�6M:<;VA+G34�8-4\Aa=>k�4�;3¥74M8�=>k

G¢ID4M8*Hr4 i ADC��B4�2�T f ¡M4�A

d(v)J34OA+G34a;<4M¥�ID4�4a=>krA+G34X�B47I�AD4*ò

vCP:

G ` A+GB6�A�CP8�¢ d(v)CK8zA+G34:B?<L�J34MI�=>k�4�;3¥74M8�67A�A+G34O�B47I�AD4*ò

vf¡M4�A

v1J34"6�:�TQ=>k�A+G34

n�B47I�ADC i 478R=>k

Gf � ?<H3Hr=B894 A+GB67A

d(v1) = mf � CK: i 4A+G34aLN6\òMCKL�?<L¿;<4M¥�ID4�4a=>ky6M:�T��B47I�AD4*òaCP8

n − 1¢M� 4aGB6��B4

m < nf�� k<� 4 i =B:38+C5;34MI

��� �

A+G34m�B47I�ADC i 478 i =B:3:<4 i AE4M;OAD=

v1JMT�6M:a4M;<¥74>¢_� 4 ¤ :<=��ÉA+GB67ArA+G34MID4 i 6M:aJ<4�:<=R4�;3¥74i =B:3:<4 i ADCK:<¥X6M:�T�H36MCPI�=>kBA+G34ML ß5=�A+G34MI���CK894B¢�� 4�� =>?325;"GB6\�B4O6XAEI�CK6M:<¥\254X6�LF=B:<¥$A+G34Aæ� =Q=>kyA+G34ML 6�:<;

v1à fX� G34M:FA+G34�:B?<L�J34MI�=>k�4�;3¥74M8���CP2@20J34�6"LF6_ò�CPLN?<L CPk�4M6 i G=>k�A+G3478-4��B47I�ADC i 478�CK8 i =B:3:<4 i AD4�;NAE=F6M2@2 A+G34"=�A+G34MI��B47I�ADC i 478O=>k

Gf � CP: i 4aA+G34MID4�6�ID4

n−m=�A+G34MI<�B47I�ADC i 478�¢\A+G34�LF6_ò�CPLN?<L ;34�¥�ID4M4$Hr=B8*8*CKJB2K4�ku=BI�4M6 i G�=>kMA+G34

m�B4MIæADC i 4M8÷�=>CK:<4M;�AE=

v1CK8

n − mfz� G>CK8$6M2P8-=�L�476�:38�A+GB67A�A+G3=B894�=�A+G347I

n − m�B4MIæADC i 4M8$6�ID4=B:B2�T�÷�=>CK:<4M;FAE=NA+G34

m�B4MIæADC i 4M8�� G>C i G�6�ID4a÷�=>CP:<4�;NAE=

v1fX� G347ID47ku=BID4>¢ C@ky� 4�GB6\�B4A+G34aLF6_ò�CPLN?<L°:B?<LFJ<47I�=>k�4M;<¥7478z��C�A+G�:<=�AEI�CK6M:<¥\25478�¢�A+G34M:

∑

v∈V

d(v) = (n − m)m + m(n − m) = 2m(n − m)f

� CK: i 4N� 4F6M2P8-=�GB6\�B4NA+G34N� 4�2P2K� ¤ :<=��a:VC5;34M:>ADC�A�T2|E| =

∑

v∈V

d(v)ß@� G347ID4

|E| CP8�A+G34":B?<L�J34MIX=>k�4�;3¥74M8Dà+¢3� 4"8-4M4 A+GB6�A |E| = m(n − m)fR� G>CP8$�>6M2@?r4"CP8O6LF6_ò�CPLN?<L 67A�A+G34 CK:>AD4�¥747Iz�>6M2@?r4�=>k

m� G>C i GFCP8X:<4M6MID4M8EA�AE= 1

2nfX� G347ID47ku=BID4>¢ C@k

nCK8$4-�B47:y¢>A+G347:m = 1

2n ` CPk n CK8$=3;3;y¢�A+G34M: m = 1

2(n − 1)

f�� G>CP8zT>C54�25;B8

|E| =

n2

4

CPknCP8$4-�B4M:y¢

n2 − 1

4

CPknCP8$=3;<; f

� =BI�AE47:NHr=>CP:>AD8\¢�A+G34aLN6\òMCKL�?<L¿:B?<LFJ<47I�=>ky8-4M¥�L�47:>AD8�CK825f

� � ��µ�� � =�894�4XA+GB6�A�A+G>CK8�LF6_ò�CPLN?<L i 6M:FJ34�6 i G>C54-�B4M;y¢M� 4�8+CKLNH32 T�A�6 ¤ 4XA+G34n�B4MIæADC i 4M8O6�:<;�8*H32@C@A�A+G34ML CK:>AD=FAæ� =Q894�A�8�=>k 1

2n�B4MIæADC i 4M8XC@k

nCK8R4-�B47:y¢0=BI$6N894�A�=>k

12(n−1)

�B4MIæADC i 4M8�6�:<; 6X8-4\A�=>k 12(n+1)

�B4MIæADC i 4M8�CPknCP8z=3;3; f0� G34M: i =B:38EAEID? i A�6M2@2Hr=B8*8*CKJB2K4"4�;3¥74M8RJ34�Aæ� 4M4M:FA+G34a�B47I�ADC i 478R=>k�;>C���4MID47:>A�894�A�8X6�:<;Q:<=Q4M;<¥7478RJ34�Aæ� 4M4M:�B4MIæADC i 4M8Rk5ID=BLÂA+G34N896ML�4N894�A f � :�¥�I�6�H<G�A+G34M=BI�Ty¢yA+G34M894F6MID4 i 6M2@254M; i =BLNH32K4�AD4�J>C5�H<6MI�ADC�AE4�¥�I�6�H<GB8 ` A+G34�¥�I�6�H<GFku=BI�AE47:��B4MIæADC i 4M8�� G>C i G�¥\C �B4M8�A+G34 LN6\òMCKL�?<L :B?<L�J34MI=>k�4�;3¥74M8�CK8$;34M:<=�AD4�;

K5,5f �

��� ���\�����U����g ��G�E�>"���C �M #C!A� 3�'$E� ��P &�(' [Q.� $@IBQ�1�q3���8>&�(�� �K�EC!�+ [DTI�"L��� $���PRQ#�3�(�P $@IL&�S ]KLS�>FG�1RSU)+&�(\"B)R*��#��� $>IL&�S ]KLS $�I�&.SU^E&�(K��I.*'�12@^E&��@��(��S (�)���FX][�U�P+������Q#P S��(7�µ ¯ ë � fEª ä � � � é�ã � ���O_rã � �7å �´�µ ° �Mä-ê� � � � ¯Mã�ä � ª ����® � �oµ

¡M4�Aa¢b¢cJ34NHr=B8+C@ADC �B4NID476M2�:B?<LFJ<47I�8O��C@A+G

a + b + c = 1f 4�C �B4M:�A+GB6�A

ab + bc + ca = 13

¢>�r:<;"A+G34X�>6M2@?r4M8$=>k�ìß56\à a

b+

b

c+

c

a

¢ ß5J�à a

b + 1+

b

c + 1+

c

a + 1

fs f � � å �BêKç � è � �c²�ç�ê ���"¯ � è Mä ����� � ± «�è ã��é7êKç � � � ±�MèBç f

� G34"¥\C �B4M: i =B:<;BC�ADC5=B:38a + b + c = 1

6M:<;ab + bc + ca = 1

3

CKLNH32 TNA+GB6�Aa = b = c = 1

3

¢3J<4 i 67?<8-4(a − b)2 + (b − c)2 + (c − a)2 = 2(a2 + b2 + c2 − ab − bc − ca)

= 2[(a + b + c)2 − 3(ab + bc + ca)]

= 2(

1 − 3 · 13

)

= 0f

�����

� G347ID47ku=BID4>¢ a

b+

b

c+

c

a= 3

6M:<; a

b + 1+

b

c + 1+

c

a + 1=

3

4

fsæs f ��� å��3êKç � è ��� � �7é7ê i¯ � ��ä�ç-± Bã�ä � ®�èBç(¯Mã�äæé*ç � � ��� � � � � ¬ �>ã�è � é ��ç5ä+ã\é ���ä �>ã�èyêKç5è� f

� ?rJ38EADC@AD?BADCP:<¥a = 1 − b − c

CP:>AE=ab + bc + ca = 1

3

¢�� 4 ¥74\A(1 − b − c)b + bc + c(1 − b − c) = 1

3

¢=BI

c2 + (b − 1)c +(

13

− b(1 − b))

= 0f

�S4 i =BLFHB?BAE4RA+G34a;BCP8 i I�CKL�CK:36M:>A�ku=BI�A+G>CK8$�B?<6�;3I�6�ADC i CK:cì

(b − 1)2 − 4(

13

− b(1 − b))

= −3(

b2 + 23b − 1

9

)

= −3(

b − 13

)2 f� CK: i 4

cCK8RID476M2�¢ � 4�ID4��>?3CKID4"A+G34F;>CK8 i IDCPLNCP:36�:>A$AE=SJ34N:<=B:<�+:<4�¥�6�ADC��B4 f�� G34MID4�ku=BID4B¢

b = 13

f � CKL�CP2P6�I�2�T ¢�� 4 i 6�:É89G3=��¶A+GB6�Ac = 1

3

6�:<;a = 1

3

f�� G347:w� 4VGB6\�B4a

b+

b

c+

c

a= 3

6�:<; a

b + 1+

b

c + 1+

c

a + 1=

3

4

f��� �������������! ��i����"L�EC%�M #C!A� 3�'$��U��P &�(� [Q�� $9I�Q�1�q#��.8#&�(��H���R���E�E���l���B�! #"�"�$[�'&�(�)R*��!,#&��U���

/'�12�3���$9����$E5K"L�@�µ ¯ ë � f[ª ä � � � é\ã�� � � �6µ ��µ � µ � Mé�êuä � �¬ �è � �å � ä+ã � uè ��ç ¨µ

� : A+G34 i =B:��B4+ò �B?<6�;3I�CP2P67AD4MI�672ABCD

¢XA+G34�;>CK6�¥7=B:36M2P8AC

6�:<;BDJBCP8-4 i AO6�:<;�AEI�CK894 i A$A+G34�=BH<H<=B8*C�AE4N6M:<¥\254786�8�8-G3=��a: f

ßK6�à � CP:<;nA+G34jßK6 i ?BAD4\à�6�:<¥\2K4jJ34�Aæ� 4M4M:AC

6M:<;BD

fßuJ�à � G3=���A+GB67A π

7< α <

3π

7

f.....................................................................................................................................................................................................

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

A B

CD

E

αα

ββ

δ

2δ

γ

2γ

��� å��3êKç � è ��� � 7é Mè �Xã�è 3ã\ä � 5é7ê �è � �7å ��² ��ä Bã � fß56\à � G34�8*?<L¿=>krA+G34a6M:<¥\25478�CP:F�B?<6�;3I�CP2P67AD4MI�672

ABCDCP8

2(α + β) + 3(γ + δ) = 2πf

� ?<LFL�CK:<¥VA+G34w6�:<¥\2K4M8FCK: 4EAB6�:<; 4ECD

TBCK472K;38α + β = 2(δ + γ)

f� ID=BL A+G34a6�J<=7�B4 4��>?<67ADCK=B:38�¢M� 4 ¥74\A

α + β =4π

7

f ß � à� 47: i 4>¢>A+G34�6 i ?BAD4a6M:<¥\254 J34�Aæ� 4M4M:

AC6�:<;

BDCK8

π − (α + β) =3π

7

f

��� �

ßuJ�à � ?<LFL�CK:<¥$A+G34O6�:<¥\2K4M8�CP: 4DAB¥\C �B4M8�A+G34R4��>?<67ADCK=B:

2α + β + δ = πf

Ú 8*CP:<¥�ß � à*¢M� 4 ¥74\Aα + δ =

3π

7

f�� CK: i 4δ > 0

¢�� 4�894�4OA+GB67Aα <

3π

7

f£ :�A+G34a=�A+G347IzGB6M:<;y¢B8+?<LFL�CK:<¥�A+G34�6M:<¥\25478$=>k 4CAB

T>C54�25;B8�A+G34a4��>?<67ADCK=B:α+2β +γ = π

f�Ú 8*CP:<¥Xß � à+¢7� 4O¥74�Aβ +γ =

3π

7` G347: i 4>¢ (4π

7− α

)

+γ =3π

7

¢=BI

γ = α − π

7

f � CP: i 4γ > 0

¢�� 4�894�4OA+GB6�Aα >

π

7

f��� ���i�����U����X ��H�������������%���3�H B"�".$K�'&�(')+*'�X,#&.����M/��12�3���$K����$�57"��98K/K 3�X5�DL��CX�3NO5���I�?3$

�M�B���1R$!,���$J57"��98@�9��C!�R 3DlI�"��E� $E��PRQ#�3�(�P $JIL&�S ]KLS9�>FG�1RSU)+&�(�"B)R*��#��� $JI�&.S ]KLS $JIL&�S ^E&�(�8!&�('�dI��2I�5kB/'<�C�����5�$�����F;m&�(�3n��P S $K�9��FZ&�(BS &3�µ ¯ ë © fEª ä � � � é�ã � ��� �6µ ��µ � µ � 7é7êuä ����¬ �è � �å � ä+ã � �è���ç�µ

Ù 4�AD4MIDLNCP:<4�A+G34�ID4M672O:B?<LFJ<47I�8a¢

b¢

c¢

d¢O¥\C��B47:�A+GB67A"A+G34�ID=3=�AD8Q=>kXA+G344��>?<67ADCK=B:

x2 + ax − b = 06�ID4

a6M:<;

c¢R6M:<;�A+G34wID=3=�AD8�=>kXA+G34�4��>?<67ADCK=B:

x2 + cx + d = 06�ID4

b6�:<;

df

� é*é�ã\è�êKç 7å@å��ê�yã�é J±�ã�é � å��3êKç � è � ����è �>ä+ã ° �Mè Mä � �0é7ê � �<ã�èyê�,¸�ç�â+ã � � âæç�ã�èyêKç��yâ �� ´�µ�²�ä � è�� � � â �>ç � � �ê�å� ���è��I²�ç�ê ���"¯ � è Mä ����� � ± «�è ã��é7êKç � � � ±�MèBç f

� CK: i 4�A+G34aID=3=�AD8�=>k A+G34 4��>?<67ADCK=B:x2 + ax − b = 0

6MID4a6�:<;

c¢�� 4aLN?<8EAGB6��B4

x2+ax−b = (x−a)(x−c) = x2−(a+c)x+ac ` A+G347ID47ku=BID4>¢ a+c = −a6�:<;ac = −b

f�� CPLNC@2K6MID2 Ty¢�8+CK: i 4XA+G34RID=3=�AD8�=>k3A+G34�4��>?<67ADCK=B:x2 + cx + d = 0

6�ID4b6�:<;

d¢M� 4O�r:<; A+GB67A

b + d = −c6�:<;

bd = df�� G>?<8\¢�� 4�=3J�AD67CK: A+G34Rku=>2@25=���CP:<¥8�T38EAE47L =>ky4��>?<67ADCK=B:387ì

a + c = −a¢

ac = −b¢

b + d = −c¢

bd = df

� kd 6= 0

¢�A+G34M:b = 1

¢�6�:<; i =B:38-4M�B?r47:>AD2�T ¢2a = −c

¢ac = −1

¢�6�:<;c + d = −1

¢�� G>C i G TBCK472K;38a = ±

√2

2

¢c = ∓

√26M:<;

d = ±√

2 − 1f

� kd = 0

¢�A+G34M:2a = −c

¢ac = −b

¢b = −c

¢k5ID=BL'� G>C i GO� 4 i 6�: i =B: i 2@?r;<4$A+GB6�A

ac = cf � 4M: i 4B¢MCPk

c 6= 0¢7A+G347:

a = 1¢b = 2

¢6�:<;c = −2

f�� =�� 4-�B4MI9¢>CPkc = 0

¢�A+G34M:a = 0

6�:<;b = 0

f� G347ID47ku=BID4>¢>A+G34�8-=>2@?BADC5=B:"CK8

(a, b, c, d) ∈{

(0, 0, 0, 0)¢(1, 2, −2, 0)

¢ (±√

2

2, 1, ∓

√2, ±

√2 − 1

)} fI�*��12i^@�12iP^@�!S (')���FX][��P������+QBP SU��(3���.QB �FJSP+P+����

��� �

µ ¯ í ± fEª ä � � � é�ã � ���O_rã � �7å �´�µ ° �Mä-ê� � � � ¯Mã�ä � ª ����® � �oµ¡M4�A

x1¢x2¢. . .¢xnJ<4$:<=B:<�*:<4M¥�67ADC �B4�ID4M672>:B?<LFJ<47I�8�896�ADCK8+k@TBCP:<¥ n

∑

i=1

xi = nf

¡M4�Axn+1 = x1

f � G3=�� A+GB6�A n∑

i=1

xixi+1 ≤ nC@k

n ∈ {1¢2¢3¢4} ¢zJB?BAa:<=�A:<4 i 4M8*896MIDC@2�T"C@k

n ≥ 5f

��� å��3êKç � è ����²\çæê ���"¯ � è Mä ����� � ± «Mèyã��é�êKç � � � ±��è>ç f�SG34M:

n = 1¢�� 4aGB6��B4

x1 = 1f�� G347:

x21 ≤ 1

f¡M4�An = 2

f$� G347:x1 + x2 = 2

f �BT�A+G34 g � 4�� � :<4M�B?<672PC�A�T ¢3� 4�=3J>A�6MCP:√

x1x2 ≤ x1 + x2

2= 1

f�� 47: i 4>¢2∑

i=1

xixi+1 = x1 + x2 ≤ 2f

� �>?<6M2@C@A�TFG3=>2K;38�CPky6M:<;F=B:B2 T�C@kx1 = x2 = 1

f� =���254\An = 3

f�� G347:x1 + x2 + x3 = 3

f �BT�A+G34X� 472@25� ¤ :<=��a:Nß56M:<;F476�8�TAE=�H3ID=M�B4_à�CK:<4M�B?<672PC�A�T3(x1x2 + x2x3 + x3x1) ≤ (x1 + x2 + x3)

2 ¢� 4a894�4OA+GB6�A

3∑

i=1

xixi+1 = x1x2 + x2x3 + x3x1 ≤ (x1 + x2 + x3)2

3= 3

f� �>?<6M2@C@A�TFG3=>2K;38�CPky6M:<;F=B:B2 T�C@k

x1 = x2 = x3 = 1f� 4+òMA i =B:38+C5;34MI

n = 4fO� G34M:

x1 + x2 + x3 + x4 = 4f �BTNA+G34 g � 4��� :<4��>?<6M2@C@A�Ty¢�� 4 ;34�;>? i 4RA+GB6�A

√x1x2 + x2x3 + x3x4 + x4x1 =

√

(x1 + x3)(x2 + x4)

≤ (x1 + x3) + (x2 + x4)

2= 2

f� 47: i 4>¢

4∑

i=1

xixi+1 = x1x2 + x2x3 + x3x4 + x4x1 ≤ 22 = 4f

� �>?<6M2@C@A�TFG3=>2K;38�CPky6M:<;F=B:B2 T�C@kx1 = x2 = x3 = x4 = 1

f� CP:36M2@2�T ¢y254\A

n ≥ 5f �S4N894�A

x1 = x2 = 12n6�:<;

x3 = · · · = xn = 0f� G347:

x1 + x2 + · · · + xn = nf � CP: i 4

n > 4¢

n∑

i=1

xixi+1 =n

2· n

2=

n

4· n >

4

4· n = n

f��� ���g���������e ��T�E�>"���C �M 3C!A9 [�'$M�U��P &�(� [Q�� $JI�Q�1�q#��.8!������������� �����! #"�"�$M�'&�(')+*'�e,#&��U���

/'�12�3���$9����$E5K"L��8�&�('�Z�9��C!�R 3DjI�"��E� $'��PRQ#�3�(�P $�I�&.S ]KLSB��FG�1RS)&�(Z"B)R*��#��� $>I�&.S ]KLS $�I�&.SU^E&�(K�

��� �

ACBEDGFIHJ>�@ D�� �~<?>�9 D�� �~<

���X,����X,��!$#�&��# ���

� 47ID4�CP8F6�H3ID=3JB2K4MLA+GB6�A�ID4��>?3CKID478Q=B:B2 T�8-=BLF4 i 6�ID4�kK?32XID476�89=B:BCK:<¥Vß56725J347C�AH<ID4\A�A�T"AEI�C i*¤ T i 6�ID4�kK?32 ID4M6M8-=B:BCP:<¥-à�6M:<;"A+G34a6�JBC@2PC�A�T�AD=�6�;3; f� �*}O��~Pï� ¿ßu��������4�I�6�;34 � 4�6M?<8*8$��=B:>AD4M8EAæà

� :RA+G34�;>CK6�¥�I�6�L�¢�A+G34$:B?<L�J34MI�8�k5ID=BL1AD=

256�ID4�AD=�J<4$6MIEI�6M:<¥74�;RCK:OA+G34

5×5¥�IDCK; 8-=OA+GB67A�476 i Ga:B?<LFJ<47I-¢�4*ò i 4MH>A16�:<;

2¢MCK8�A+G34�8*?<L =>k>Aæ� = =>kBC�AD8�:<4�C5¥7G3J3=>?<I�8 fß � ?<L�J34MI�8�CK:�A+G34S¥�I�C5;w6MID4 è ã_ç ��� � � ��äæé CPkzA+G34�CKI8-�>?<6�ID478�AE=>? i G�6725=B:<¥R6a8*CK;<4a=BI�67A�6 i =BIE:<47I f�� =BI4*òM6MLFHB254>¢XA+G34 �

1��GB6M8

8:<47CK¥7G3J<=>?<I�8 f à � =BL�4=>kOA+G34w:B?<LFJ<47I�8SGB6\�B4w6M2PID4M6�;�T�J34�47:n�<2@254M;jCP: f

�SG>C i G�:B?<L�J34MI�LN?<8EA�ID47H32P6 i 4XA+G34�� � �a� G34M: A+G34¥�IDCK;�CK8 i =BLFHB254\AE4M; �

20 216 5 4

23 7 1 3 �9 8 2

25 24 22

� G>CP8�CK8a:<=�AO6�:<=�A+G347I � ?r;3= ¤ ?��jG3=B:<478+A � � AX25=3= ¤ 8a6QJBC�AO2@C ¤ 4�=B:<4>¢0A+G3=>?r¥7G f� GB6�A�CK8�H<6MI�Az=>krA+G34�ID4M6M8-=B: � G�T�A+G>CK8�H<ID=3J>2547L �a6M8�CP: i 2P?r;34�;N=B:"A+G34a�y=B:>AE478+A��ÉC�ACK8z:BC i 4�AD="GB6\�B4OH<ID=3J>2547LF8�A+GB67A�25=3= ¤ k56�L�CP2@CK6MI�J>?BA-¢�?<H<=B: i 25=B894MI�4*òM6MLNCP:367ADCK=B:y¢�6�ID46�J>C@A�;>C���4MID47:>A f��� å��3êKç � è � �S4 i =>?325;�÷5?<8+AX�r;<;>254F6MID=>?<:<;�JMT�AEIDCP6M2�6M:<;�4MIDID=BIX?<:>ADC@20� 4F¥74�AO8-=BLF4:B?<L�J34MI�8�A+GB67A�� =BI ¤ f �<?BA�� 4X��C@2P2B�a672 ¤ A+GBID=>?r¥7G�A+G34a8-=>2@?BADC5=B:"CP:N6a2K=3¥\C i 6M2B�a6\T f

� A-§ 8�AE=>?r¥7GRAE= ¤ :<=��Ü4+òM6 i AD2 TO� G34MID4$AD=a8EAD6MI�A f�� CPI�8+A-¢7C@A0LF6 ¤ 4M8�8-47:38-4�AD= i G34 i*¤� G>C i GV:B?<LFJ<47I�8 6�ID4FLNCP898+CK:<¥ f�� G34Q¥�I�C5;V6M2PID4M6�;�TSCP: i 2P?r;34M8RA+G34�:B?<LFJ<47I�81AE=

96�:<;20AE=

25 ` 89= A+G3=B8-4�LNCP898+CK:<¥O6�ID4 10AD=

19f

� 4+òMA-¢_� 4 i =>?32K;O� ¥\?<ID4$=>?BAr� G>C i GR:B?<LFJ<47I�8�CP:OA+G34$¥�I�C5;R6�ID4�672KID476�;�TXA+G34�8*?<L=>k Aæ� =N:<4�C5¥7G3J3=>?<I�8 f � =BI�4+òM6�LNH32K4B¢9GB6M8�:<4�C5¥7G3J3=>?<I�8

16�:<;

8ß56M:<;

9 = 1 + 8à `

8GB6�8R:<47CK¥7G3J<=>?<I�8

16M:<;

7ß56M:<;

8 = 1 + 7à+¢�6�:<;S89=S=B: f ¡M4\A-§ 8OC@A�6M2@C i C�Þ*4N4-�B47I�T:B?<L�J34MIR� G>C i G�CK8�6M2PID4M6�;�T�A+G34�8+?<L =>kzAæ� =w=>k�C@A�8":<47CK¥7G3J<=>?<I�8\¢�6M8a� 4�2P2z6�8 A+G344M:>AEI�C5478

16�:<;

2f

20 21� � �23 � � � �� � �25 24 22

� =���� GB6�A � � A-§ 8$H<ID=3JB6�J>2�T�ADCPL�4�ku=BI�A+GB67A�AEI�C54M;F6M:<;�AEID?r4aH<ID=3J>2547L��+8-=>2 �BCP:<¥AE4 i GB:BC5�>?r4���H36�:BC i�f0g kPAE47I�� 4�¥74�A�A+GB67Az=>?BAz=>ky=>?<I�8�TB8+AD4ML�¢M� 4�L�C5¥7G�A�AEI�T"25=3= ¤ CK:<¥67A089=BL�4$=>k�A+G34$:B?<L�J34MI�8�A+GB6�A�GB6��B4$6M2PL�=B8EA�6M2@2>=>k�A+G347CPI0:<4�C5¥7G3J3=>?<I�8�6M2PID4M6�;�TR�32P2K4�;CK: f�g 2K89= ¢M� 4�LNCK¥7G�A�6M8�� 472@2<ku= i ?<8$=B:"A+G34aH<6MI�A�=>k<A+G34 ¥�IDCK;N:<476�I�A+G34 � � � f

� � �

� =BI�4+òM6�LNH32K4B¢ i =B:38*CK;<47I21f � CK: i 4

21672KID476�;�TaGB6M8�:<47CK¥7G3J<=>?<I�8

206�:<;

4¢_� 4LN?<8EA��aIDC�AE4

216M8�47C�A+G34MI

20 + 1=BI

4 + 17f �<?BA�A+G34R:B?<L�J34MI

1672KID476�;�T�6MH<H<4M6MI�8472P8-4_� G34MID4RCK:"A+G34a¥�IDCK; ` A+G>?<8\¢�A+G34 47LFH>A�TN8*H<6 i 4aJ<4�25=�� 21

L�?<8+A�J3417f

20� �� � �17

23 � � � �� � �25 24 22

¡M=3= ¤ CP:<¥ 6�A176�8�� 4�;BCK;N��C�A+G

21¢B� 4�894�4aA+GB6�A

17L�?<8+A$J<4

3 + 14=BI

4 + 13 `A+G>?<8�¢<A+G34 � � ��L�?<8+A$ID47H<ID478-47:>A�47C�A+G34MI 13=BI

14f �<?BAz� 4 i 6�:y§ A$896\TFku=BI$8+?<ID4aTB4\A� G>C i GN=B:<4RC@A�CP8 f

� =�� 6�J3=>?BA22 � � A i 6�:3:<=�AaJ34

2 + 20¢�6M8

20CP8 6M2PID4M6�;�T�6 i+i =>?<:>AE4M;�ku=BI f

�SGB67AyAæ� =a:B?<LFJ<47I�8�6�;<;RAE=226M:<; 6MID4�:<=�AyTB4\A0CK:RA+G34X¥�IDCK; � � G34O=B:B2 TaHr=B8*8*CKJBC@2PC�A�TCK8

106�:<;

12¢0CK:V8-=BLF4�=BID;34MI f �<?BA i 6M:�� 4"AE4�2P2�� G>C i GV=>k

106�:<;

12CK8�HB2K6 i 4M;� G34MID4 � � k 10 �a6M8�6�J<=7�B4 22

¢_� 4 i =>?32K;R:<=�A�¥74�A106M8�A+G34�8*?<L =>kMAæ� =O:<4�C5¥7G3J3=>?<I�8�¢8*CP: i 4

2 + 86M:<;

3 + 76MID4 :<=�A�Hr=B8*8*CKJB2K4 f � k

12CK8$6�J<=7�B4

22¢BA+G34M:

12 = 10 + 26�:<;10 = 8 + 2

¢�� G>C i G i 6M:�� =BI ¤rf20

� �� � �17

23 � � � �� � � � �25 24 ��� � �

�S4 ¤ :<=��hA+GB6�A�A+G34�� � ��CK8z47C�A+G34MI 13 =BI 14 f �y=>?325; C@A�J<4 13 � g ID4�A+G34MID4XAæ� =:<47CK¥7G3J<=>?<I�8X=>k � � �FA+GB6�A�6�;3;�AD= 13 � � = f � ="A+G34�� � ��LN?<8EA�J<4 14¢�� G>C i GN89=>2��B478A+G34aH<ID=3J>2547L f�<?BA��a6MC�A � �S4 i 6�:y§ Az8+AD=BHF:<=�� ��¡M4\A-§ 8 i 6MIEI�TF=B:N6 J>C@A�kK?<IæA+G34MI f¡M=3= ¤ CP:<¥Q6�A

25¢�� 4S894�4FA+GB67A

25L�?<8+A�J34

24 + 1ßK:<=�A 6SHr=B8*8*CKJBC@2PC�A�T�àR=BI

9 + 16f � 47: i 4>¢�A+G34�:B?<L�J34MIRCP:VA+G34�8*H<6 i 4�6�J<=7�B4

25L�?<8+A�J<4

16fS� G>CP8 :<=��6M2@25=��a8z?<8�AD="C@A�6M2@C i C�Þ*4

23¢24¢25¢36M:<;

16f ß �SG�T � à� � � �� � � ���� � � � � � �

� � � � � � �� � � � ��� � �

� I�T i =BLNH32K4�ADCP:<¥XA+G34aID478+A�=>k<A+G34 ¥�IDCK;F=B:�TB=>?<Iz=��a: �

� ���

� ;��~H ����:�H�� ���G>�B,K �JD��?K �1BED?@ � ��B�@L:����UK� �`<?DGDIH�� ��B�H ACBEDGFIHJ>�@

��� �X#��X� � ��(

g k56�LF=>?<8�4*òM6MLFHB254�=>k 6aID4 i ID4M6�ADC5=B:3672rLN67A+G347LF6�ADC i 8�H<ID=3J>2547L CK8 �yCKI ¤ LN6�:y§ 8��i G3=3=>25¥\CPID2�1<ID=3JB2K4ML � ��� f�� G>CK8zH3ID=3JB2K4ML�¢M� G>C i G�;B67AD4M8�k5ID=BL A+G34RL�C5;3�*:BCP:<4�AD4�47:>A+Gi 47:>AD?<I�Ty¢ i 6�:FJ34aH<GBI�6�894�;N6M8�ku=>2@25=��a8�ì

g 8 i G3=3=>2K�EAD4M6 i G34MIy��CK89G34M8�AD=�H3ID=M�>C5;34O6��a6M2 ¤ CP:<¥�8 i G34M;B?32K4Xku=BI15¥\CPID2P88-=QA+GB6�A-¢�=7�B4MIO6�H<4MI�C5=3;�=>kz6�� 4M4 ¤ ¢�4M6 i G�¥\CPID20�a6M2 ¤ 8a;367CP2 TQ��C@A+GQAæ� =i =BLNH<6M:BC5=B:380JB?BA :<=�Aæ� =X¥\CKI�2K8�6�H3Hr476�I<AD=3¥74�A+G347IrCK:O6zA+GBID4M4M89=BL�4�Aæ��C i 4 f�y6M:�TB=>?�G3472PH�A+G34RAD4M6 i G34MI��r:<;N8+? i G�6 8 i G34�;>?3254 �g :�TS8+? i GS8 i G34�;>?3254���C@2P2�672P2K=��°4M6 i GV¥\CPID2�AE=F�a672 ¤ ��C�A+G

14;>C���4MID47:>A i =BL��H<6M:BC5=B:38O=7�B4MIzA+G34 i =>?<I�8-4"=>k06�� 4�4 ¤ f � :�=�A+G347I�� =BID;38\¢ 4M6 i G�¥\CKI�2r��CP2@2<�a6M2 ¤ ��C�A+G4-�B4MI�TF=�A+G347I�¥\CPID2 4*òM6 i AD2�TN=B: i 4 f

� A�;<=3478�:<=�A�A�6 ¤ 4Q25=B:<¥FAD=�;BCP8 i =M�B47I�A+GB6�Aa�r:<;>CK:<¥�8+? i Gw6V8 i G34�;>?3254�CP8":<=�A6�AEIDC �BCP6M2�LN67A�AD4MI f �BT�AEI�CK672�6�:<;É4MIDID=BI-¢���C�A+GwHr47I�8-4-�B47I�6�: i 4B¢$=B:<4�LF6\T�Hr=B8*8*CKJB2 TJ<4�8*? iEi 47898+kK?32 fag :<=�A+G347IX6MH<H3ID=B6 i GQCK8XAD=Q�<:<;S6M:S6�H3H<ID=BH3IDCP67AD4F¥74M:<47I�6M2@C@Þ+6�ADC5=B:S=>kA+G34VH<ID=3J>2547L 6M:<;�;>CK8 i =7�B4MIaCK:38+C5¥7G�A�8FJMT�8-=>2 �BCP:<¥�8+CKLNH32K4MI i 6M8-478 fn� G>CP8�L�4\A+G3=3;8-=BLF4�ADCPL�478�25476�;B8zAE=�;BCP8+AEI�6 i ADC5=B:38 f¡M4�Ay?<8�2K=3= ¤ 6�AyA+G34 i 6�894z� G34MID4z� 4z�>6�I�TOA+G34�:B?<LFJ<47I�=>k3¥\CPID2P8�6�:<;�6�;�÷5?<8+AyA+G34:B?<L�J34MIO=>k�;B6�TB8OAD=S47:38*?<ID4"A+GB6�AO4M6 i GS¥\CKI�2y�a6M2 ¤ 8���C�A+GS4-�B47I�TV=�A+G34MIX¥\CKI�2�4*òM6 i AD2�T=B: i 4 fg 8 i G3=3=>25�DAE476 i G347I���CK89G34M8�AE=FH3ID=M�>C5;34�6��a6M2 ¤ CP:<¥a8 i G34M;B?32K4 ku=BI

v¥\CPID2P88-=RA+GB6�A-¢>=7�B4MI�6OHr47IDCK=3;"=>k

(v −1)/2;36\T38\¢>476 i G�¥\CPID2��a672 ¤ 8z;367CP2 TR��C@A+GAæ� = i =BLNH<6M:BC5=B:38"JB?BA :<=VAæ� =w¥\CKI�2K8��a6M2 ¤ AD=3¥74�A+G347I�CK:w6QA+GBID4�478-=BLF4Aæ��C i 4 f � =BI�� GB67A��>672P?r478$=>k

vi 6M:"A+G>CK8$J<4a;<=B:<4 �

� =BIv = 9

¢rA+G347ID4"CP8O6N¥74M=BL�4\AEIDC i �a6�T�=>ky�>CK8+?<6M2@C@ÞECP:<¥ 6�8-=>2@?BADC5=B: f"Ù 47:<=�AE4A+G34"¥\CPID2P8XJMT�A+G34�:B?<LFJ<47I�81A+GBID=>?r¥7G

9f � H3ID4M6�;NA+G34�:B?<LFJ<47I�8

1AE=

8?<:BCPku=BIDLN2 T6M2K=B:<¥RA+G34 i CKI i ?<L�ku4MID47: i 4"=>k06 i CPI i 254 6M:<;F2P6�J3472rA+G34 i 4M:>AEID4"=>k A+G34 i CKI i 2K4 6�8

9f � :A+G34 i CKI i 2K4B¢<;3I�6\� A+G34�AEI�CK6M:<¥\25478

(2, 7, 8)6�:<;

(3, 4, 6)6M:<;�A+G34�;BCP6�¥7=B:3672

(1, 9, 5)ßPA+G34�;BCP6�¥7=B:3672 i 6�:�J<4a�>C54_� 4�;�6�8X6N;34�¥747:<4MI�6�AE4 AEIDCP6�:<¥\2K4\à fO� G>CP8R¥\C��B478O6a�a6M2 ¤ CP:<¥

.

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................

..............................................................................................................................................................................................................................................................................

......................................................................................................................

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..................................................................................................................................................................................................................................................................................

............................................................ ................................................................................................................................................................................................................................................................................

..

.

..

..

..

..

.

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

q

q

q

q

q

q

q

q

q

12

3

45

6

7

8

9

�������

.

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................

..............................................................................................................................................................................................................................................................................

......................................................................................................................

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..................................................................................................................................................................................................................................................................................

............................................................ ................................................................................................................................................................................................................................................................................

..

.

..

..

..

..

.

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

q

q

q

q

q

q

q

q

q

1

2

34

5

6

78

9

�������

.

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................

..............................................................................................................................................................................................................................................................................

......................................................................................................................

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..................................................................................................................................................................................................................................................................................

............................................................ ................................................................................................................................................................................................................................................................................

..

.

..

..

..

..

.

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.q

q

q

q

q

q

q

q

q 1

23

4

5

67

8

9

�������

.

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

..

..

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...........................................................

..............................................................................................................................................................................................................................................................................

......................................................................................................................

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..................................................................................................................................................................................................................................................................................

............................................................ ................................................................................................................................................................................................................................................................................

..

.

..

..

..

..

.

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

q

q

q

q

q

q

q

qq

12

3

4

56

7

89

������� �"!$#&%(' )+*+,

c© -/."."0 �21/3&1$4"' 1/365718,8*"9+:;18,<' =>1/?2@/�&=<' 9<,(#

� � �

6�IDI�6�:<¥747L�47:>AOku=BI�A+G34��rI�8EAO;36\T f �BTSID=�A�67ADCP:<¥�A+G34 i CPI i 254>¢45;<4M¥�ID4�478a6�AX6�ADCPL�4>¢JB?BA ¤ 4�47H3CP:<¥$A+G34$AEIDCP6�:<¥\2K4M8��>ò�4M;y¢�� 4R=3J�AD67CK:�6$�a672 ¤ CK:<¥�8 i G34�;>?3254�� G34MID4R476 i G�¥\CPID2�a6M2 ¤ 8���C�A+G�Aæ� =N;>C���4MID47:>A i =BLNH<6M:BC5=B:38�ku=BI�476 i GN=>k

48*? iEi 47898+C��B4a;B6�TB8 f�<?r=MTB4M;jJMTwA+G>CP8N8+? i+i 4M8*8�¢$=B:<4V8*H<ID476�;B8"A+G34V:B?<L�J34MI�8

1AD=

146M2K=B:<¥QA+G34i CKI i ?<L�ku4MID47: i 4�=>k�6 i CKI i 2K4B¢�2P6�J3472P8"A+G34 i 4M:>AEID4�=>k�A+G34 i CPI i 254

156�:<;�;BI�6��a8"A+G34;BCP6�¥7=B:3672

(1, 15, 8)f"� G34N:<4+òMAOG3=>?<IXCP8 i G34�47IDkK?32@2�TS89H<4M:>AXHB2K6 i CK:<¥�ku=>?<I$AEIDCP6�:<¥\2K4M8��C@A+G>CP:�A+G34 i CPI i 254�6�:<;�6MH<HB2�T>CK:<¥NA+G34�ID=�AD6�ADC5=B:3672�AD4 i GB:BCK�B?r4N� G>C i G�� =BI ¤ 4M;S� 472@2ku=BI

v = 9f � ID?<8EAEI�67ADCK=B:S8*CP: ¤ 8OCP:S6�8R:<=�89=>2P?BADCK=B:QCK8Oku=BIæA+G i =BL�CK:<¥ f � CP:36M2@2�T ¢0=B:<425=3= ¤ 8X67A�A+G34 i 6M8-4

v = 9AE=F8-4M4a� GB6�A �+� 4M:>A$I�C5¥7G�A ��6M:<;Q6�:3672�T�Þ*4M8O=B:<4>§ 8XID4 i 4M:>A8 i I�C5J3JB2@CK:<¥RAE=N8-4M4�� GB6�A �+� 4M:>A��aID=B:<¥ �Oku=BI

15f�� G34�H3ID=3JB2K4ML CP8$254�kPA�6M2K=B:<4aku=BI�6;36\TF=BI�Aæ� = ¢�A+G347:F6a8*?r;3;<47:NCK;<476a2K4M6�;38zAD=�6a8-=>2@?BADC5=B:"ku=BI

v = 27f

.

..

.

..

..

.

..

..

.

..

..

..

.

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

..

.

..

.

..

.

..

..

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................................................................................................................

.........................................................................................................................................................................................................................................................................

...

....

...

...

....

...

...

....

..

..

..

...

..

..

..

...

..

..

..

...

..

.

..

..

.

..

..

.

..

..

..

.

..

..

.

..

..

.

..

..

..

.

..

..

.

..

..

.

..

..

..

.

..

..

..

..

...

..

..

..

...

..

..

..

...

..

....

...

...

...

....

...

...

....

..........................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

....

...

....

...

....

...

....

...

...

.............................................................................................................................

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

pq

q

q

q

q

q

q

q

q

q

q

qq

q

q

q

q

q

q

q

q

q

q

q

q

q

1 23

4

5

6

7

8

9

10

1112

13141516

17

18

19

20

21

22

23

2425

26

27

� G34 ;>CK6�¥�I�6�L¿89G3=��a8�H<6MI�A�=>krA+G34��<I�8+A�;B6�T § 8$6MIEI�6M:<¥74MLF4M:>A ` :36�LF472 T"A+G34 ;>C5�6�¥7=B:3672(1, 27, 14)

6�:<;�6 i =B:<¥�ID?r47:>A0H<67CKI�=>k�AEI�CK6M:<¥\25478(5, 13, 15)

6�:<;(2, 26, 18)

f�BTN6�;3;BCP:<¥OA+GBID4M4aH<67CKI�8�=>k i =B:<¥�ID?r47:>A�� Aæ��CP:y§�AEIDCP6�:<¥\2K4M8(3, 4, 9)

6�:<;(16, 17, 22)

¢(6, 10, 25)

6M:<;(12, 19, 23)

¢É6�8�� 4�2P2V6�8(7, 21, 24)

6�:<;(8, 11, 20)

¢�=B:<4i =BLNH32K4�AD4M8OA+G34N6�IDI�6�:<¥747L�47:>AOku=BI$A+G34N�<I�8+AO;36\T fI7 =�AD6�ADCK:<¥�A+G34N:B?<L�J34MI�8 =B:�A+G34i CKI i 2K4B¢r=B:<4�8EAE47HQ6�A�6aADCKLF4B¢r¥\C��B478$A+G34 6�IDI�6�:<¥747L�47:>AD8�ku=BI�A+G34�:<4*ò7A12;B6�TB8 f�� G348*CK¥�:BCP� i 6M:>AXku4767AD?<ID4N=>k�A+G>CK8R6�IDI�6�:<¥747L�47:>AXCK8XA+GB6�AR476 i G�H36MCPIO=>k i =B:<¥�ID?r47:>A�AEIDCP6�:<�¥\25478 i 6�:�J34V6M8989= i CK6�AE4M;�AD=�6�AEIDCPH32K4�A {a

¢b¢

c = a + b} ¢�� G34MID4 a¢

b¢�6�:<;

c6�ID4XA+G34R2547:<¥_A+GB8$=>k3A+G34�8-G3=BIæAE478+A�6�I i 8$J34�Aæ� 4M4M:�A+G34��B4MIæADC i 4M8�=>k<A+G34OAEI�CK6M:<¥\254 f�� G34AEIDCP6�:<¥\2K4(2, 26, 18)

i =BIEID4789H<=B:<;38jAD= {2¢

8¢

10} ` (3, 4, 9)i =BIEID4789H<=B:<;38jAD=

{1¢5¢6} ` (10, 6, 25)

i =BIEID4789H<=B:<;38$AE= {4¢7¢11} ` 6M:<; (21, 24, 7)

i =BIDID4M8*Hr=B:<;B8AE= {3¢9¢12} f0� G34M894�ku=>?<I AEI�CKHB254\AD80ku=BIEL'6�H36�IæADC@ADCK=B:a=>k�A+G34$894�A {1

¢2¢. . .¢12} f � ACK8z:<=�A�H<=B898+C5J>254�AD= �r:<;�8*? i G�6�H36�IæADC@ADCK=B:�=>k {1

¢2¢3¢4¢5¢6} ` A+G>?<8\¢�A+G>CP8zL�4\A+G3=3;i 6M:3:<=�A�J34�?<894�;�AE=�89=>2��B4��yCKI ¤ LN6�:y§ 8�=BI�C5¥\CP:36M2rH<ID=3J>2547L f

�SGB67A�GB6��B4a� 4�¥�6MCP:<4�; � ��=BID4"H<ID=3J>2547LF8�A+GB6M:�6�:38D� 4MI�8\¢yJB?BA�� 4�;3=�GB6\�B46 8+AEI�6�AE4M¥-T"AD=N89=>2��B4��yCKI ¤ LN6�:y§ 8 ��i G3=3=>2K¥\CKI�2�1<ID=3J>2547L°ku=BI i 47I�A�6MCP: i 6�894M8 f � 47ID4a6�ID48-=BLF4�:<4\� i GB6M2@2547:<¥74M8zA+GB6�A�TB=>?FLF6\T�2@C ¤ 4OAE= AD6 i*¤ 2K4OTB=>?<I�89472@k ft �>ô7~�~@ï�îi��ï�� f � G3=��¿A+GB6�A�C@kzA+G34Q894�Aa=>kzCP:>AE4M¥74MI�8

A = {1¢2¢

. . .¢3n} i 6M:�J<4H<6MI�ADC�ADC5=B:<4M;�CK:>AD=

n;>CK8D÷�=>CK:>AX8+?rJ3894�A�8

Ai = {ai¢bi¢ci}

ßPku=BIi = 1

¢2¢. . .¢nà8*? i GaA+GB67A

ai + bi = ciku=BI�4M6 i G

i¢MA+G34M:�47C�A+G34MI

n = 4k=BI

n = 4k + 1ku=BI�8-=BLF4CK:>AD4�¥747I

kf

� ���

��¤ =>2547L � � � GB6M8X8-G3=��a:NA+GB6�A�C@kn = 4k

=BIn = 4k + 1

ku=BI�89=BL�4aCK:>AD4�¥747Ik¢�A+G347:�8*? i G�6�H36�IæADC@ADCK=B:N=>k

A = {1¢2¢. . .¢3n} 4+ò�CP8+A�8 f ��=>?<I�:<4*ò7A i GB672P2K4M:<¥74OCP8L�=BID4�L�=3;34M8EA�A+GB6�:"A+G>CP8 f

t �>ô7~�~@ï�îi��ï�� f � CP:<;�672P2�H<6MI�ADC�ADC5=B:38 =>k�A+G34�894�AR=>k�CP:>AE4M¥74MI�8A = {1

¢2¢. . .¢3n}CK:>AD=

n;BCP8E÷�=>CP:>AR8*?rJB8-4\AD8

Ai = {ai¢bi¢ci}

ßPku=BIi = 1

¢2¢. . .¢nàE¢�8+? i GSA+GB6�A

ai + bi = ciku=BIz4M6 i G

if�Ù =�A+G>CP8�ku=BI

n = 4¢5¢8¢B6�:<;

9f

t �>ô7~�~@ï�îi��ï�� f�� CK:<;�6aH<6MI�ADC�ADC5=B:�k5ID=BL¿�yGB6M2@2547:<¥74��OAE="8-=>2 �B4���CPI ¤ LF6M:y§ 8 ��i G3=3=>25¥\CPID21<ID=3JB2K4ML f�Ù =�A+G>CK8�ku=BIv = 27

¢33¢51¢36M:<;

57f

��GB672P2K4M:<¥74 � i 6�:VJ<4F�>?3C@AD4�k5ID?<8EAEI�67ADCP:<¥�¢08*CP: i 4N:<=�AX672P2�H<6MI�ADC�ADC5=B:38O2K4M6�;QAE=�68-=>2@?BADC5=B:FAD=NA+G34 ��i G3=3=>2K¥\CKI�2z1<ID=3J>2547L f � 4M: i 4B¢rC@k�=B:<4"H<6MI�ADC�ADC5=B:�CP8$AEID=>?rJB2K4M89=BL�4>¢A+G34M:"AEI�TN6M:<=�A+G34MI f�� G34MID4O��C@2P2rJ<4a6�A�2K4M6M8+Az=B:<4aH36�IæADC@ADCK=B:"A+GB6�A�� =BI ¤ 8 �� CP:36M2@2�T ¢�� 4�8-G3=>?32K;N:<=�A�ku=BID¥74\A�A+G34aLF=�ADC��>6�ADC5=B:"ku=BI�A+G>CP8�6�IæADC i 2K43ì

t �>ô7~�~@ï�îi��ïO� f�� =>2��B4��yCKI ¤ LN6�:y§ 8�=BIDCK¥\CK:3672rH<ID=3J>2547L f� =BH<47kK?32@2�TFTB=>? i 6M:��r:<;�8-=BLF4�ID4-�B=>2P?BADCK=B:36�I�TQC5;34M6�� G>C i G i 6M:VJ<4"?<8-4M;QAD=8-=>2 �B4a=�A+G34MI i 6M8-478 � � =���¢3G3=�� 6�J<=>?BAz6MIEI�6M:<¥\CK:<¥OA+G3=B894a¥\CKI�2K8�CP:F¥�ID=>?<H<8�=>krku=>?<I �t }" � Nï�îMõ f g :R4+ò�H<=B8*C�AE=BI�TX6�IæADC i 2K4B¢ � ç5ä O±�Mè�· é � â � � � å �7çKä�å Mè ����ã-åBê�ã � ª ä � �Må�ã�±Né ¢i 6M:�J34zku=>?<:<;XCK:�A+G34X�w6\T �M� � ��4M;BC�ADC5=B:R=>k ��i CK4M:>ADC@� i<g LF4MI�C i 6�:RßK8-4M4�672K89= ��� � à f � =BI6aL�=BID4�;<4\AD67CP2K4�;N;>CK8 i ?<898+C5=B:�=>k i T i 2@C i 6M2<8-=>2@?BADC5=B:38\¢>894�4�� � � f,7 6�T3�9�yGB6M?r;3G>?<IDC 6M:<;

�QCP2P8-=B: � � � 8-=>2 �B4�;�A+G34 ¥74M:<47I�6M2@C@Þ*4M; ��CPI ¤ LF6M:y§ 8 ��i G3=3=>2K¥\CKI�2�1<ID=3J>2547L JMTN8-G3=���CP:<¥A+GB67Az6a8 i G34M;B?32K4 4+ò�CP8+A�8�ku=BI�4-�B47I�TF=3;<;�LN?32�ADCKHB254a=>k3f

� ï_óæï �*ï�î �_ï7ö� � � � f � f&7�f �r672P2�6�:<; �Nf �<f � f ��=\ò�4\AE47I-¢��yCKI ¤ LN6�:y§ 8 ��i G3=3=>2K�R4�CKI�2K8�1<ID=3JB2K4ML�¢° 3ê� ã�±�BêKç�â 7å���ã�âDä+ã BêKç � èBé Mè � � é*é ��Mé ¢ ������� 4M; f�� 4\� ��=BI ¤ ì��w6 i L�CP2@2K6M:�y= f ¢ ��� ��� f

�D� � � f 4�6MID;3:<47I-¢ Ù CK:3:<47I�4�?r478+A�8�¢ ��i G3=3=>2K¥\CKI�2K8�6M:<; � 6M:<; i ? ��4�;�1<IDCP8-=B:<47I�8�¢ ² � ã¸�7é7ê���ãMâDä*ã 3êKç � è3é ¢ � H3IDCP:<¥74MIE�D¦�47ID2P6�¥ � 4_� ��=BI ¤ ¢ �M��� � f

� � � �yf 1 f ��CPI ¤ LF6M:y¢�?r47I�T�¦ � ¢ ¸� ����· é �è�� ��ã�èyêKå�ã�±��è�· é��zçMä � ß � � � ��àE¢3H f � � f� ��� ÙRf � f�7 6\T<�*��GB67?r;<G>?<I�Cy6�:<; 7zf � f �QCP2P8-=B:y¢ � =>2P?BADCK=B:38�=>k �yCKI ¤ LN6�:y§ 8 ��i G3=3=>2K�4�CKI�2y1<ID=3JB2K4ML�¢ �,±�ã�ä)µ ° 3ê�1µ ��� âdµ � �J±���µ ª �Mä*ã ° 3ê�1µ � �Rß ��� ��� à+¢ � � � �M� � f

� � � � G f ��¤ =>2547LS¢ £ :w��47I�A�6MCP: Ù CP8+AEI�C5J>?BADC5=B:38�=>k � :>AE4M¥74MI�8 CK:j1>6MCPI�8���C�A+GG4�C �B4M:Ù C �047ID4M: i 4M8\¢ ° Bê��µ � â �è���µ�� ß �M� ��� àE¢ � � � � fg LN6�I � =3;3G>CÙ 4MH36�IæAEL�47:>A�=>k��w6�A+G34MLN67ADC i 8�6M:<; � AD6�ADCK8EADC i 8

� CPI �QC@2Pk5ID4M;[4�ID47:Bku472@2y��=>2@254M¥74�y=BIE:<47IX�rID=3= ¤ ¢ � ¡>¢3�y6M:36�;B6 g � � �<1 �sodhi@swgc.mun.ca

� � �

����� � G � � ����� � ����� ���� �����

��������� ��� ��#3����S4QJ<4M¥\CK:SA+G>CK8 :B?<LFJ<47IX��C�A+G�H3ID=3JB2K4MLN8ak5ID=BL A+G34FAæ� =�;B6�TB8�=>k�A+G34 � CP:36M2

7 =>?<:<;�=>kBA+G34 ��� ��� ��=BID4M6M:��w6�A+G34MLN67ADC i 672 £ 2 T3LNH3CP6�;R�aIDC�A�AE47: CK: g H3IDC@2r�M��� � f �VTA+GB6�: ¤ 8�¥7=�AD=R��GBI�CK8EAE=BH<G34MI � LF672P2�¢M� 6�:36�;BCP6�: � 476�L ¡M4M6�;<47IrAE=�A+G34 � � £ CK: g A+G34M:38\¢ku=BI i =>2P2K4 i ADCP:<¥XA+G347L°ku=BI�=>?<I�?<8-4 fØ ������������� ��!�µ"�$# %&��µ"��# 'M¨(�*)+��)-,jµ/.0'1�*2È�ÊuÅ�ËrÌ Ä ����Å�Æ

m�3 � ��~ �5476 � �98��74:4��Ø f�� G34aCK: i CPI i 254

O=>k�6�:NCK89=B8 i 472K4M8zAEIDCP6�:<¥\2K4

ABC��C@A+G

AB = ACLF4�4\AD8

BC¢

CA¢�6M:<;

AB6�A

K¢L¢�6�:<;

M¢�ID4789H<4 i ADC �B472 T f ¡M4\A

NJ34�A+G34XCP:>AE47I�8-4 i ADC5=B:"=>kBA+G342PCP:<4M8

OL6�:<;

KM¢r6M:<;F254\A

QJ34aA+G34 CK:>AD4MI�894 i ADCK=B:Q=>kyA+G34 2PCP:<4M8

BN6M:<;

CAf¡M4�A

PJ<4NA+G34Nku=3=�A�=>k�A+G34�H<4MIDHr47:<;BC i ?32K6MIRk5ID=BL

AAE=

BQf � k�� 4F6�8*8*?<LF4NA+GB6�A

BP = AP + 2PQ¢�� GB67Az6MID4RA+G34�Hr=B8*8*CKJB2K4X�>6M2@?r4M8$=>k

AB/BC �¯ f�� G3=���A+GB67Az:<=�H36MCPI�=>kyH<=B8*C�ADC��B4OCK:>AD4�¥747I�8x6M:<;

y896�ADCK8+� 478

3y2 = x4 + xf

á f�gji =BLNH3?BAD4MI�:<4�Aæ� =BI ¤ CK8�ku=BIELF4�;�JMT i =B:3:<4 i ADCK:<¥2004

i =BLNH3?BAD4MI�8XJMT i 6�JB2K4M8 fg 894�AS=>k3A+G34M894

2004i =BLFHB?BAE47I�8�CP8�8967C5; AD=�J34 ç5è��<ã ��ã\è �<ã\è�ê CPkr:<="H36MCPI�=>k i =BL��H3?BAD4MI�8$=>k

SCK8 i =B:3:<4 i AD4�;NJMT"6 i 6�J>254 f�� ?<H<H<=B8-4XA+GB67A�A+G34R:B?<L�J34MIz=>k i 6�J>25478z?<8-4M;CK8�A+G34$L�CK:BCPLN?<L�?<:<;<47IyA+G34 i =B:<;>C@ADCK=B:OA+GB67A A+G34�8*C�Þ*4$=>k>6M:�TRCP:<;<47Hr47:<;<47:>A08-4\A0;3=34M8:<=�A�4*ò i 4�4M;50f

ßK6�à Ù 4M:<=�AD4"JMTc(L)

A+G34":B?<LFJ<47I�=>k i 6�JB2K4M8�?<894�;NAD= i =B:3:<4 i AzA+G34 i =BLNH3?BAD4MIL��C@A+GF=�A+G347I i =BLNH3?BAD4MI�8 f � G3=��'A+GB67A-¢3ku=BIz6�:�TFH<67CKI�=>k i =BLFHB?BAE47I�8

A6�:<;

B¢

c(A) = c(B)C@k0A+G34-TQ6�ID4 i =B:3:<4 i AD4�;�JMT�6 i 6�JB2K4"6�:<; |c(A) − c(B)| ≤ 1=�A+G34MI���CK894 f

ßuJ�à � CP:<;"A+G34�:B?<L�J34MI�=>k i 6�J>25478�?<894�;"ku=BI�A+G34�:<4�Aæ� =BI ¤ fë f�� G34MID4�6MID4

nHr=>CP:>AD8R=B:�6 i CKI i 2K4B¢ :B?<L�J34MID4M;Qk5ID=BL

1AE=

nf ¡M4�A

SJ<4 A+G34"8-4\A=>k0A+G3478-4"H<=>CK:>A�8 f ¡M4\A G J<4 A+G34�k56�L�CP2 T�=>k�672P2 k 472K4MLF4M:>A$8*?rJB8-4\AD8 A=>k

S� G>C i GGB6��B4�A+G34 H<ID=BH<4MIæA�T�A+GB67A-¢rJ34�Aæ� 4M4M:F6�:�T�Aæ� =F;>CK8EADCK: i A�Hr=>CP:>AD8

i6�:<;

jCP:

A¢BA+G347ID46�ID4�67A�2K4M6M8+A

3H<=>CK:>A�8�=>k

S� G>C i G�6MID4a:<=�A�CP:

Af

� =BIn¢k ≥ 2

¢>�<:<;"A+G34a:B?<LFJ<47I�=>k�472K4MLF4M:>A�8�=>k G fí f ¡M4�AR6�:<;

rJ<4�A+G34 i CPI i ?<LFI�6�;BC@?<8�6M:<;RA+G34�CP:3I�6�;>CP?<8�=>k 4ABC

¢7ID4M8*Hr4 i ADC��B4�2�T f� ?<H<H<=B8-4OA+GB67A

∠ACK8�A+G34R2K6MID¥74M8EAz=>k<A+G34OA+GBID4�4�6�:<¥\2K4M8�=>k 4ABC

f ¡M4\AMJ<4OA+G34LNCK;<�+Hr=>CP:>A�=>k

BC6�:<;

XJ34�A+G34zCK:>AD4MI�894 i ADCK=B:�=>k7A+G34�AD6M:<¥74M:>A�8�AD=$A+G34 i CPI i ?<L i CPI i 254=>k 4ABC

6�AB6M:<;

Cf�� G3=���A+GB6�A

r

R≥ AM

AX

f

� � �

² f ¡M4�ApJ<4a6aH3IDCPL�4>¢B6�:<;F;347�<:<4

fp(x) = xp−1 + xp−2 + · · · + x + 1f

ßK6�à � G3=���A+GB6�A-¢�ku=BIy6M:�T�H<=B8*C�ADC��B4zCK:>AD4�¥747Im;>C��>CK8+C5J>254�JMT

p¢\A+G347ID4�4+ò�CP8+A�8�6$H3IDCPL�4

qA+GB67A�;>C��>C5;34M8

fp(m)6M:<;�CK8�ID4�2K6�ADC��B4�2�T�H3IDCPL�4OAE=

m(m − 1)f

ßuJ�à � G3=��nA+GB6�A�A+G34MID4X6MID4�CP:B�r:BC�AE4�2�TaLF6M:�T H<=B8*C�ADC��B4$CP:>AE4M¥74MI�8n8*? i GRA+GB6�A

pn+1CK8�6aH3IDCPL�4 f

� 4+òMA�� 4�¥\C��B4�A+G34 H<ID=3J>2547LF8X=>k A+G34"� ��� � �r6M2 ¤ 6�:S�w67A+G347LF6�ADC i 6M2 £ 2 T3LNH3CP6�;����� � f�� GB6M: ¤ 8$6�¥�6MCP:F¥7=�AD=N��GBI�CK8EAE=BH<G34MI � LN6M2@2<ku=BI i =>2P2K4 i ADCP:<¥XA+G347L f¯ Ø�� � à �*) � ��! µ"��# % �"µ"�$# '7¨(�*) ��)-,�µ/. ' �*2 �Ì�Ç<Ô�Ç ÅÜ�y�0� �

Ø f�� G34a894��>?r4M: i 4a0¢a1¢a2¢. . .

8*67ADCP8*�r4M8�A+G34aID4�2K6�ADC5=B:am+n + am−n − m + n − 1 = 1

2(a2m + a2n)ku=BIz672P2r:<=B:<�+:<4�¥�6�ADC��B4�CP:>AE4M¥74MI�8

m6M:<;

n��C@A+G

m ≥ nf�� k

a1 = 3¢>�r:<;

a2004f

¯ f�� CP:<;N672P2rH3IDCPL�4a:B?<LFJ<47Iz8-=>2@?BADC5=B:38�AE=�A+G34a4��>?<67ADCK=B:xy − yx = xy2 − 19

fá f ¡M4�A

OJ<4�A+G34 i CKI i ?<L i 47:>AEID4�=>kBA+G34O6 i ?BAE4�AEI�CK6M:<¥\254

ABCf0� G34 i CPI i 25478 i 4M:>AEID4M;67A�A+G34OLNCK;<�+Hr=>CP:>AD8�=>k3A+G34�AEIDCP6�:<¥\2K4B§ 8�8+C5;34M8z6M:<;"H<6M898+CK:<¥�A+GBID=>?r¥7G

OCK:>AD4MI�894 i A�=B:<46�:<=�A+G347Iz67A�A+G34aH<=>CK:>A�8

K¢L¢36M:<;

Mf 1<ID=7�B4RA+GB6�A

OCK8zA+G34�CP: i 47:>AEID4�=>krAEI�CK6M:<¥\254

KLMf

ë fV� G34QHB2K6M:<4�CP8�;>C��>C5;34�;VCK:>AD=�H<6MI�A�8�JMT�6��r:BC�AE4�:B?<LFJ<47Ia=>kz2@CK:<478 fS� � =�H36�IæAD86�ID4 i 672P2K4�; �-:<47CK¥7G3J<=>?<I�CK:<¥ ��C@k3A+G34-TNGB6\�B4�6 i =BLNL�=B:�894�¥�LF4M:>A-¢>GB6M2@ku�+2@CK:<4>¢B=BI�2@CK:<4 f� 6 i G�H<6MI�A�CK8zAD=NJ34a6M898+C5¥�:<4M;N6aID4M672r:B?<L�J34MIz8*? i G�A+GB6�A_ìßPC�à�A+G34RH<ID=3;>? i Az=>k<A+G34R6M898+C5¥�:<4M;":B?<L�J34MI�8zCK:�4M6 i G"H36MCPI�=>k :<4�C5¥7G3J3=>?<IDCP:<¥RH36�IæAD8CK8�2K4M8*8zA+GB6M:�A+G34�8*?<L¿=>krA+G3=B894RAæ� =":B?<L�J34MI�8 `ßKC@C�à�A+G34a8+?<L =>k�672P2r:B?<LFJ<47I�8�6M898+C5¥�:<4M;"AE=�H36�IæAD8�2 TBCP:<¥�=B:�A+G34a8*6�LF4a8+C5;34 =>ky6M:�T=>krA+G34a¥\C��B47:N2@CK:<478�CK8$4M�B?<672<AD=

0f

1<ID=M�B4OA+GB67A�A+G>CP8�CP8�Hr=B8*8*CKJB2K4RC@ky6�:<;F=B:B2 T"CPkrA+G34R2PCP:<4M8�6MID4a:<=�Az672P2rH36�I�672P2K472 f

� G34zID4MLN6MCP:BCK:<¥�894�A<ku=BIBTB=>?<I H3ID=3JB2K4MLÜH32K4M6M8*?<ID478�6�ID4�k5ID=BL�A+G34 ��� ��� �*6�H36�:<478-4�w67A+G347LF6�ADC i 6M2 £ 2 T3LNH3CP6�; ¢ � CP:36M2yID=>?<:<; ¢B�aIDC�A�AE47: � 4�JBID?<6MI�T��M��� � f�� GB6�: ¤ 8X6�¥�6MCP:¥7=�AE=N�yGBIDCP8+AD=BHrG347I � LN6M2@2rku=BIz=3J>A�6MCP:BCK:<¥XA+G347L°ku=BIz=>?<I�?<894 fØ ë ����� �*. ��! � ��� µ"�$# %&��µ"��# 'M¨(�*)+��)-,jµ/.0'1�*2È0ÊuÅ�Ë Ì Ä ����Å�ÆzÓ�È�Ç���Ï���Ë Ï ���0��Ó��y�0� � � ÊuÖjÇ � ��� ����Ï����

Ø f 1<ID=7�B4OA+GB6�A�A+G34MID4OCK8�:<="H<=B8*C�ADC��B4XCK:>AD4�¥747Inku=BI0� G>C i G

2n2 + 1¢3n2 + 1

¢>6M:<;6n2 + 1

6�ID4a672P2rH<4MI�ku4 i Az89�B?<6MID4M8 f

� � �

¯ f � CP:<;V6M2@2�kK?<: i ADCK=B:38f : � → �

8*? i GQA+GB67A-¢0ku=BIO6M2@2�ID476M2�:B?<LFJ<47I�8x6�:<;

y¢

f(

xf(x) + f(y))

=(

f(x))2

+ yf

á f ¡M4\AABC

J34�6�AEI�CK6M:<¥\254 f ¡M4�ASJ<4 A+G34 i CPI i 254 A+GB6�A$CK8OH<4MIDHr47:<;BC i ?32K6MI�AE=FA+G34H32P6�:<4

ABC6�:<; GB6M8�;>CK6ML�4\AE47I

ABf ¡M4�A

D6�:<;

EJ<4�H<=>CK:>A�8�=B:

S¢M6�:<;a8+?<H<H<=B8-4A+GB67A

DEL�4M4�A�8

AB67A

Pf � G3=�� A+GB67AXCPk

CPJBCP8-4 i AD8

∠ACB¢�A+G347:

CP6M2P8-=JBCP8-4 i AD8

∠DCEf

ë f�� =BI�Hr=B8+C@ADC �B4�ID476M2r:B?<L�J34MI�8a¢b¢36M:<;

c��C@A+G

a + b + c = 1¢B8-G3=���A+GB6�A

1 + a

1 − a+

1 + b

1 − b+

1 + c

1 − c≤ 2

(

b

a+

c

b+

a

c

) f��=>?F:<4M4�;N:<=�Az8EAD6�AE4X� G347:�4M�B?<672PC�A�TNG3=>25;B8 fí f�£ :�A+G34$CP8*2P6�:<;O� G>C i G���C ¤ 6$AD=>?<ID4�;a2P6�8EA TB4M6MI-¢�4M6 i GO�BC@2P2P6�¥74�CK8�2@CK: ¤ 4M;�JMTa6XID=B6�;AE=V4+òM6 i AD2 TQA+GBID4�4F=�A+G34MI��BC@2P2P6�¥7478 f � =�8-G34N8-4\AR=>?BA�k5ID=BL¾6"�>CP2@2K6�¥74B¢ �>CK8+C@AD4�;V4-�B47I�T�BC@2P2P6�¥74a4+òM6 i AD2 TF=B: i 4B¢B6M:<;NID4\AD?<IE:<4M;�AD=�A+G34��<I�8+A��BC@2P2P6�¥74 f� G>CP8aTB476�I9¢z89G34QCP8�H32P6�:3:BCP:<¥FAE=w¥7=w=B:w6�ID=>?<:<;�AEIDCPH�6�¥�67CK:y¢�8EAD6MI�ADCP:<¥�k5ID=BLA+G34�896ML�4��>CP2@2K6�¥74B¢7�BCP8*C�ADCK:<¥O4-�B4MI�T �>CP2@2K6�¥74�4*òM6 i AD2�T�=B: i 4�6M:<;�ID4�AD?<ID:BCK:<¥XAD= A+G34O�rI�8EA�BC@2P2P6�¥74 f �<?BA�8-G34O;<=3478�:<=�A��a6�:>A0AD=aku=>2P2K=��nA+G34XCK;<47:>ADC i 6M2 i =>?<I�894O6M8�2K6M8+AyTB4M6MI-¢�:<=BI÷5?<8+A0A+G34OID4-�B4MI�894OID=>?BAE4 f�� G3=���A+GB67A0A+G34MID4XCP8z6M:�C�ADCK:<47I�6�I�T�A+GB67A�LF6 ¤ 4M8zG34MI�GB6�H3H�T f

� : Ù 4 i 4MLFJ<47IN�������B¢z� G34M:w� 4�¥�6\�B4�ID476�;34MI�8\§�89=>2P?BADCK=B:38"AD=ÉH3ID=3JB2K4MLN8F=>kA+G34 £$i AD=3J<47Iz����� � � � äEè ã�ä ¢7� 4RLNCP89894�; i C�ADCK:<¥$A+G34R8-=>2@?BADC5=B:�=>k0��C i G34�2��r6�AD67CP2@254�AD=H<ID=3J>2547L � =>kyA+G34 � 4 i =B:<; 7 =>?<:<;�=>k�A+G34������>� ��?r¥7=B8+2K6\���?<672PC@� i 67ADCK=B:�ku=BI � � £�����>�Q¥\C��B47:�CK: ������� � ì � �M� � � � ¢z�M�����wì � � � � f �S4F6M2P8-=V=M�B47ID2K=3= ¤ 4�;É�r67A�6MC@2P2K4B§ 88-=>2@?BADC5=B:�AE=OH3ID=3JB2K4ML � =>kMA+G34�¦�CK:<¥_A+� � 4MH>ADCDÝ47L�4 £ 2�TBLFHBCK6�;<4X�w6�A+G��4MLN67ADCK�B?r4R� 4�25¥74ßæ��CK;BC � CP:36M2K4\ày� G>C i G�6MH<H<4M6MID4�;�CP: ������� � ì � � � � � � ` �������Nì � � � � f �VT�6MHr=>2K=3¥\C5478AE=Q��C i G3472��r67A�6MC@2P2K4 f

� 4+òMAQ� 4�H<ID478-47:>A�6h89=>2P?BADCK=B:�AE='H3ID=3JB2K4ML�� �h=>k�A+G34 � ��� �w6 i 4M;<=B:BCP6�:�w67A+G347LF6�ADC i 6M2 £ 2 T3LNH3CP6�; f�� G>CP8�89=>2P?BADCK=B:RCK8�89G3=BI�AD4MI A+GB6M:RA+G34�=B:<4zA+GB67Ar�a6�8�¥\C �B4M:CK: ���������Fì>� � � � �M� � f¯ f�Ù =34M8zA+G347ID4 4*òMCK8EA�6akK?<: i ADCK=B:f : � → �

8*? i G�A+GB67A�ku=BIz4-�B47I�Tn ≥ 2

¢f(

f(n − 1))

= f(n + 1) − f(n) �

��� å��3êKç � è ���[¸�ç�� � � ��� ª � å � � ±�± ��è>çæê � � � å@å�ã �Bã ���RçKè�ê�ã�ä � O¯Mã�è ���¨¸ ��® � �oµ� ?<H<H<=B8-4RA+GB6�A

fCK8�8+? i GN6�kK?<: i ADCK=B: f�� G34M:y¢>ku=BIz672P2

n ≥ 2¢

f(n + 1) − f(n) = f(

f(n − 1))

≥ 1f

� ���

� G>?<8\¢fCK8�CP: i ID4M6M8*CP:<¥Rku=BI

n ≥ 2f �BT�CP:<;B? i ADCK=B:y¢�� 4 GB6��B4

f(n) ≥ n − 1ku=BIz6M2@2

n ≥ 2f� k

f(8) ≤ 9¢�A+G34M:

3 = 9 − 6 ≥ f(8) − f(7) = f(

f(6))

≥ f(5) ≥ 4¢

� G>C i G"CK8�6 i =B:>AEI�6�;>C i ADCK=B: f � 4M: i 4B¢f(8) ≥ 10

f�� G347:f(10) > f(10) − f(9) = f

(

f(8))

≥ f(10)¢

� G>C i G"CK8�6�¥�6MCP:N6 i =B:>AEI�6�;BC i ADC5=B: f

� 4+òMA�� 4�AD?<IE:�AE=j8-=>2@?BADC5=B:38NAD=nH3ID=3JB2K4MLN8�=>kXA+G34n�rIDC�ADCK89GÜ�w67A+G347LF6�ADC i 6M2£ 2�TBLFHBCK6�;��M������������� � ¢ 7 =>?<:<; � ¢3¥\C��B47:N67A����������Fì>� � ��� fØ f 4�C��B47:SA+GB67A

34! = 295 232 799 cd9 603 140 847 618 609 643 5ab 000 000¢;<4\AE47IEL�CK:<4RA+G34a;BCK¥\C@A�8

a¢b¢c¢df

��� å��3êKç � è ��� ª ç�ã�äEä+ã�¬ � äEè3é��\ê�ã-ç5è � ° �çKé � è3é�³R¸� � ê�ê�ã ����ä��è â+ã"µ� =BI$6�:�T�H<=B8*C�ADC��B4�CP:>AE4M¥74MI

n6�:<;�6M:�T�H3IDCPL�4

p¢ C�A$CK8$� 4�2P2 ¤ :<=��a:QA+GB6�A$A+G344*ò�Hr=B:<47:>A�=>k

pCK:"A+G34�H<I�CKLF4 ;<4 i =BLNHr=B8+C@ADCK=B:F=>k

n!CK8

νp(n!) =∑

k≥1

⌊

n

pk

⌋ ¢� G34MID4 b.c ;34M:<=�AD4M8$A+G34 CK:>AD4�¥747I$H36�IæA f � 47: i 4>¢ ν2(34!) = 32

6M:<;ν5(34!) = 7

f� Azku=>2@25=��a8�A+GB6�A

34!CP8$;BC �BCP8*CKJB2K4aJMT

107 ¢�� G>C i G"ku=BI i 4M8 b = 0f

� CK: i 4ν2(34!) = 32

¢<C�A�ku=>2@25=��a8�A+GB6�A34!/107 CP8X;>C��>CK8+C5J>254 JMT 8

f�� G347:FA+G343�*;BCK¥\C@Aa:B?<L�J34MI

35a� G3=B8-4�;BCK¥\C@A�8"6MID4�A+G34

3I�C5¥7G�A+�+L�=B8EA ;BCK¥\C@A�8�=>k

34!/107 CP8;BC �BCP8*CKJB2K4XJMT8f0� G>?<8\¢

300+50+aCK8�;BC �BCP8*CKJB2K4XJMT

8f0� G347:

a+6 ≡ 0 (mod 8)¢��C@A+G

6 ≤ a + 6 ≤ 15f�� G34MID4�ku=BID4B¢

a + 6 = 86�:<;

a = 2f

� =BI�6�:�T�Hr=B8+C@ADC �B4RCK:>AD4�¥747In¢>254\A

n =k∑

i=0

ai10i J34RA+G34a;<4 i CKLN6M2 4*ò�H<6M:38*CK=B:

=>kn¢r6M:<;F254\A

Se(n) =∑

i≥0

a2i6M:<;

So(n) =∑

i≥0

a2i+1f � A�CP8�� 472@2 ¤ :<=��a:FA+GB6�A

n ≡ Se(n) + So(n) (mod 9)6�:<;

n ≡ Se(n) − So(n) (mod 11)f � ID=BL A+G34¥\C��B47: ;<4 i CKLN6M2�4*ò�H<6M:38*CK=B:Rku=BI

34!¢\� 4��<:<;OA+GB67A

Se(34!)+So(34!) = c+d+1406�:<;Se(34!) − So(34!) = c − d − 18

f�£ :"A+G34a=�A+G34MIzGB6�:<; ¢B8*CP: i 496�:<;

116�ID4J<=�A+Gak56 i AD=BI�8�=>k

34!¢M� 4OGB6��B4

34! ≡ 0 (mod 9)6M:<;

34! ≡ 0 (mod 11)f0� G>?<8\¢

c + d + 140 ≡ 0 (mod 9)6�:<;

c − d − 18 ≡ 0 (mod 11)f�� G>CP8a2K4M6�;38RAE=

c + d ≡ 4 (mod 9)6�:<;

c − d ≡ 7 (mod 11)f � CK: i 4

0 ≤ c + d ≤ 186M:<;

−9 ≤ c − d ≤ 9¢BC�A�ku=>2@25=��a8�A+GB67A

c + d = 4=BI

c + d = 13¢

6�:<;c − d = −4

=BIc − d = 7

f� =���¢�8*CP: i 4

c − d6M:<;

c + dGB6\�B4NA+G34F896ML�4NH<6MIDC�A�TV6�:<;

c ∈ {0¢1¢. . .¢9} ¢� 4 476�8+CP2 TN;<4M;B? i 4RA+GB67A

c = 06�:<;

d = 4f� G347:

(a, b, c, d) = (2, 0, 0, 4)f

� � �

� ï� �ô���� f�� G34aH3ID=3JB2K4ML°8+A�67AD4MLF4M:>A�CP8$4MIDID=B:<4�=>?<8 f �w6MH32K4 ¥\C �B4M834! = 295 232 799 cd9 604 140 847 618 609 643 5ab 000 000

f� G34QID4M6M8-=B:BCP:<¥�6�J3=M�B4 i 6M:�J<4Q6�;36MHBAD4�;V� =BID;�ku=BIX� =BID;VAE=�H<ID=7�B4�A+GB6�A-¢�CK:�A+GB6�Ai 6M8-4>¢

(a, b, c, d) = (2, 0, 0, 3)f

¯ f0� G34$AEIDCP6�:<¥\2K4ABC

¢�� G347ID4AB < AC

¢�GB6M8 i CKI i ?<L i CKI i 2K4Sf�� G34XH<4MIDHr47:<;BC i �?32K6MI0k5ID=BL

AAE=

BCL�4M4�A�8

S6�¥�6MCP: 67A

Pf0� G34XHr=>CP:>A

X2PCK4M8�=B:�A+G34�2@CK:<4�8-4M¥�L�47:>A

AC¢B6�:<;

BXL�4M4�A�8

S6�¥�6MCP:N67A

Qf

� G3=���A+GB67ABX = CX

C@ky6�:<;N=B:B2�T"CPkPQ

CP8�6�;>CK6ML�4\AE47I�=>kSf

��� å(¯Mã�� � � ��ã ��� ä+ã � �oµ ��Mè � �åPå � � J± �<ã�è � �1² � ® � � � �è�� ��µ ¶Jµ � ±�ã�ã\è� ����åæê � � ±�±�ã-å ��ê�yã5´aã�ê� ã�ä�å �è���éJµ �"ã��7ç(¯Mã ��Mè � �åPåw· é$é � å �BêKç � è1µ

� ?<H<H<=B8-4RA+GB6�ABX = CX

f0� G347:∠XCB = ∠XBC

= ∠QBC = ∠QAC `� G34M: i 4B¢

AQ ‖ BCf � A�ku=>2P2K=��a8�A+GB67A

∠PAQ = 90◦ ¢B� G>C i G�L�476�:38$A+GB6�A PQCK8�6 ;BCP6�LF4�AD4MIz=>kSf� G34a6MID¥\?<L�47:>AzCP8�ID4-�B4MI�8+C5J>254 f

..

.

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

.

..

..

.

..

.

..

.

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................

..

..

..

..

...

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

..

...

..

..

..

..

..

..

..

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............................

................................

................................

..........

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

.

..

.

.

..

..

.

.

..

........................................... .

.

.

.

.

.

.

.

..

..

....

..

..

..

..

..

.

..

.

.

.

..

.

..

..

.

..

..

..

...........

A

B CD

P

Q

X

á f ¡M4\Ax¢y¢zJ34�H<=B8*C�ADC��B4aID4M672y:B?<L�J34MI�8X8+? i G�A+GB67A

x2 + y2 + z2 = 1f 1<ID=M�B4A+GB67A

x2yz + xy2z + xyz2 ≤ 13

f��� å(¯Mã�� ��� ° ç�â �yã-å�¬ 3ê 7ç@åPå�ã � � � �>ã�è � ��ä MèyâEã�� ª ç�ã\äEä*ã ¬ � äEèBé �\ê�ã_çKè�� ° 7çPé � èBé3³¸� � ê�ê�ã � ��ä MèyâEã��3¶ � é�«ã�¸ �7çPé �t«� ��³/¬ �äEä+ã�ä � ��®0è>ç$¯Mã\ä�é+çæê 3ê ª � åPç�ê��ã�âDè>ç�â �<ã ��Bê�å �Mè � �¬ �ä+â+ã-å � è��� � ��7çKè � ��ã ��� ä*ã ���oµ ��Mè � �åPå � � J± �<ã�è ���1² �E® � � �X_rã � �7å 8´�µ ° ��ä-ê ���� � ¯Mã�ä � ª ��� ® � � � Mè � � �¨­ Mä � ²Jµ �8µ � �è ��� �RçPå � ä�ç � ¸� �Mä�ç�ã\ä ®0è>ç$¯Mã\ä�é+çæê ���� 3ê�ã�ä�å � � � � ´�µ �"ã �<ä�é�ê �7ç(¯Mã�ê�yãXé � å��3êKç � è ��� ¬ Bê�çPå@å�ã"µ

� ID=BL A+G34 � 6M? i G�T ��i G��a6MI�Þ � :<4M�B?<672PC�A�T�6�:<;"A+G34aG�T3H<=�A+G34M8+CK8\¢�� 4aGB6��B4x2yz + xy2z + xyz2 = xyz(x + y + z)

≤ xyz(12 + 12 + 12)1

2 (x2 + y2 + z2)1

2 =√

3 xyzf

£ :QA+G34F=�A+G347IXGB6�:<; ¢ 13

= 13(x2 + y2 + z2) ≥ 3

√

x2y2z2¢y?<8+CK:<¥ A+G34 g � 4��� :<4��>?<6M2@C@A�T ` G347: i 4>¢ xyz ≤ 1

3√

3

fX� G34�;<478*CPID4�;FCK:<4M�B?<672PC�A�TFku=>2P2K=��a8�CPLFLF4�;>CK6�AE4�2�TJMT i =BLFJBCP:BCK:<¥XA+G34RAæ� ="ID4M8+?32@A�8 f´aã <ê�­Oã���ç$¯Mã ê�yãOé � å �BêKç � è ��� �o«� ��³R¬ MäEä*ã\ä � µ

� ID=BL A+G34�CK;<47:>ADC@A�T(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

6�:<;A+G34O� 472@25� ¤ :<=��a:"CK:<4M�B?<672PC�A�Ta2 + b2 + c2 ≥ ab + bc + ca

¢>C�A�ku=>2@25=��a8�A+GB67A3(ab + bc + ca) ≤ (a + b + c)2

f ß � à

� �M�

� 4�A�ADCP:<¥a = xy

¢b = yz

¢B6�:<;c = zx

CP:Fß � à�TBCK472K;383(x2yz + xy2z + xyz2) ≤ (xy + yz + zx)2 ≤ (x2 + y2 + z2)2 = 1

¢k5ID=BL � G>C i G � 4a¥74\A

x2yz + xy2z + xyz2 ≤ 13

f� =�ADC i 4OA+GB67A�4M�B?<672PC�A�TNG3=>25;B8�� G34M:x = y = z = 1√

3

fë f ¡M4�A

m6M:<;

nJ34 CP:>AE4M¥74MI�8O¥�ID4767AD4MI�A+GB6�:

1f ��=B:38+C5;34MI�6�:

m × nID4 i AD6M:<¥\?32K6MI¥�IDCK;Q=>k0H<=>CK:>A�8�CP:NA+G34�HB2K6M:<4 f � =BLF4

k=>kyA+G3478-4 Hr=>CP:>AD8�6MID4 i =>25=>?<ID4M;�ID4M;FCK:F8*? i G6X�a6�TaA+GB67A�:<= A+GBID4M4�ID4M;"Hr=>CP:>AD8�6MID4XA+G34��B4MIæADC i 4M8�=>k 6RIDCK¥7G�A+�*6M:<¥\254M;�AEI�CK6M:<¥\254XAæ� ==>k�� G3=B894"8+C5;34M8X6�ID4"H36�I�672P2K472 AE=NA+G34"8+C5;34M8O=>k�A+G34�¥�IDCK; f�Ù 4\AE47IEL�CK:<4 A+G34�¥�ID4767AD4M8EAHr=B8*8*CKJB2K4X�>6M2@?r4 =>k

kf

��� å��3êKç � è ��� ª ç�ã�äEä+ã�¬ � äEè3é��\ê�ã-ç5è � ° �çKé � è3é�³R¸� � ê�ê�ã ����ä��è â+ã"µ�S4���C@2P2yH3ID=M�B4>¢yJMTFCP:<;B? i ADCK=B:�=B:

p = m + n¢3A+GB6�A�A+G34�¥�ID4M6�AE478+A�Hr=B8*8*CKJB2K4�>6M2@?r4 =>k

kCK8

m + n − 2f

trôMö�ï � fp = 4

f� G34�¥�IDCK;�CK82 × 2

¢�6�:<;VA+G34QLF6_ò�CPLN?<L :B?<LFJ<47I =>k�ID4�;�Hr=>CP:>AD8 CP8 i 2K4M6MID2 T2 = 2 + 2 − 2

f�� G>?<8\¢�A+G34aID4M8+?32@AzG3=>25;B8�ku=BIp = 2

ftrôMö�ï�� f

p ≥ 4fg 898+?<L�4 A+GB67A�A+G34�ID478*?32�AXG3=>25;B8Oku=BI$6M2@2

m¢n > 1

8+? i G�A+GB6�Ap = m + n

f¡M4�A�?<8 i =B:38+C5;34MI�6 ¥�IDCK;���C@A+Gp + 1 = m + n

f�QC@A+GQ:<=�2K=B898R=>k�¥747:<4MI�672PC�A�T ¢<� 4�LN6�T�6M898+?<L�4 A+GB6�A$A+G34�¥�IDCK;QCK8Xku=BIELF4�;�JMT

mID=��a8a6M:<;

ni =>2P?<LN:38R��C�A+G

m ≥ nfQ� G347:

m ≥ 3f ��=B:38+C5;34MIO6 i =>2K=>?<IDCP:<¥��C@A+G�:<=�ID4M;"I�C5¥7G�A+�+6�:<¥\2K4�; AEI�CK6M:<¥\254�Aæ� =�=>kB� G3=B8-4O8+C5;34M8�6�ID4OH<6MI�6M2@254�2>AE=aA+G34O8*CK;<478=>k0A+G34N¥�I�C5;�ß56�J<JBID4-�BCP67AD4�;�:<=QID4M; 7<g0� à f � CK: i 4

m ≥ n¢ CPk�4M6 i G�2PCP:<4 i =B:>AD67CK:38O6�A25476�8EA�Aæ� =aID4M; Hr=>CP:>AD8\¢7A+G347:�6�A025476�8EA�Aæ� = =>k>A+G347L'� =>?32K;�GB6\�B4X6OID4M; Hr=>CP:>A�=B:aA+G34896ML�4 i =>2@?<LF:y¢�� G>C i G � =>?325;N¥\C��B4�6 ID4M; 7<g0� ¢B6 i =B:>AEI�6�;BC i ADC5=B: f� G>?<8\¢MA+G347ID4O4*òMCK8EAD8�6�2PCP:<4���C�A+Ga:<=aL�=BID4�A+GB6�:�=B:<4XID4M; Hr=>CP:>A f�Ù 472K4�ADCP:<¥�A+G>CK82PCP:<4B¢y� 4�=3J�AD67CK:V6 i =>25=>?<ID4M;

(m − 1) × nID4 i AD6M:<¥\?32K6MI�¥�I�C5;Q��C@A+GS:<=SID4�; 7rg0� ¢6�:<; ��C�A+G

p = (m − 1) + nf�� G34RCK:<;>? i ADC5=B:NG�TBHr=�A+G3478*CP8$47:38*?<ID478zA+GB67A�A+G347ID4�6�ID467AOL�=B8EA

(m − 1) + n − 2ID4�;VHr=>CP:>AD8�CP:SA+G>CP8 ¥�IDCK; f�� G>?<8�¢�A+G34MID4F6�ID4F6�ARLF=B8+A

(m − 1) + n − 2 + 1 = m + n − 2ID4�;NH<=>CK:>A�8�CK:"A+G34�CP:BC@ADCP6M2r¥�IDCK; f��=B:��B47I�8-4�2�T ¢3JMT i =>25=>?<I�CK:<¥OID4M;�6M2@23A+G34RHr=>CP:>AD8zCK:�A+G34R�rI�8EA�ID=��'6M:<;"A+G34O�rI�8EAi =>2@?<LF:�4+ò i 47HBA�A+G34�Hr=>CP:>Az� G>C i GFCP8�CP:�J<=�A+GNA+G34 �rI�8EA$ID=�� 6�:<;NA+G34��<I�8+A i =>2@?<LF:y¢� 4 ¥74\Az6 i =>25=>?<I�CK:<¥���C�A+GN4+òM6 i AD2 T

m + n − 2ID4M;NH<=>CK:>A�8$6M:<;N:<=�ID4M; 7rg0�yf� G347ID47ku=BID4$A+G34OLN6\òMCKL�?<L :B?<L�J34MI�=>k<ID4M;�H<=>CK:>A�8�CK8

m+n−2¢�� G>C i G 4M:<;B8A+G34�CK:<;>? i ADC5=B:N6M:<;"A+G34aH3ID=3=>k f

í f�� CP:<;N672P2r89=>2P?BADCK=B:38�CK:NH<=B8*C�ADC��B4OCK:>AD4�¥747I�8a¢b¢cAE= A+G34 4��>?<67ADCK=B:

a!b! = a! + b! + c!

��� å(¯Mã�� ��� ° � ��X±�±�ã � � Mé*é*ç@å � � êuä�Mé�� � ��ä ��� ��ä��è â+ã � � � � � ���è � ��è ��¬ � ä �<ã � ���ä��è â+ã � �è�� ª ç�ã�äEä+ã ¬ � äEèBé �\ê�ã_çKè�� ° 7çPé � èBé3³/¸� � ê�ê�ã � ��ä MèyâEãOµ �"ã �7ç(¯Mã ê� ãé � å��3êKç � è ��� � 7é9é+çPå ¨µ

� ka 6= b

¢�A+G347:V��C�A+G3=>?BA�2K=B898�=>k�¥747:<4MI�672PC�A�T ¢a > b

f�� G34M:a!;3=34M8 :<=�A;BC �BCK;<4

b!f � 4M: i 4B¢

c! = a!b! − a! − b!CK8�:<=�Az;BC �BCP8*CKJB2K4aJMT

a!f�� G34M:

a > c¢>6�:<;

���M�

a!b! = a! + b! + c! < 2a! + b!¢ � G>C i G�CPLFHB2PCK4M8XA+GB6�A

(a! − 1)(b! − 2) < 2f

� AzCP8$4M6M8�T�AE= i G34 i*¤ A+GB6�A�A+G>CK8$¥\C �B4M8�:<=�8-=>2@?BADC5=B:38 f� G347ID47ku=BID4>¢M� 4XLN?<8EA�GB6\�B4a = b

f�£ ?<I�4��>?<67ADCK=B:�J34 i =BL�478a!2 = 2a!+ c! `G34M: i 4

a! | c!6M:<;�A+G>?<8

a < cf ��IDC�AE4

c = a + k� G347ID4

k > 0CK8X6�:QCP:>AE4M¥74MI fÙ C��>CK8+C5=B:�JMT

a!:<=��hTBCK472K;38

a! = 2 + (a + 1)(a + 2) · · · (a + k)f�� G347:

a! > 2¢� G>C i G"CKLNH32@C5478zA+GB6�A

a > 26M:<;

3 | a!f�� 47: i 4>¢�� 4�L�?<8+A�GB6\�B4

(a + 1)(a + 2) · · · (a + k) ≡ 1 (mod 3)f

� G347:k < 3

ßu=�A+G347Iæ��CP8-4(a + 1)(a + 2) · · · (a + k)

CP8�;>C��>CK8+C5J>254RJMT3à f�� k

k = 1¢� 4"¥74�A

a! = a + 3¢3� G>C i G�TBCK472K;38

a = 3 ` A+G347: (a, b, c) = (3, 3, 4)f�� k

k = 2¢� 4�¥74�A

a! = a2 + 3a + 4¢>� G>C i GNCPLFHB2PCK4M8�A+GB67A

a | 4 ` G347: i 4>¢ a = 1¢a = 2

¢r=BIa = 4

f�� A�CK8$476�8�T�AD= i G34 i*¤ A+GB67A�A+G3478-4 i 6M8-478�k56MC@2 f� G>?<8\¢>A+G34a=B:B2�T�89=>2P?BADCK=B:�CP8(a, b, c) = (3, 3, 4)

f

� =�� � 4�2K=3= ¤ 6�AO89=>2P?BADCK=B:38Ok5ID=BL¾=>?<IXID4M6�;<47I�8RAD= 7 =>?<:<;��F=>k�A+G34S�rI�C@ADCP8-G�w67A+G347LF6�ADC i 6M2 £ 2�TBLFHBCK6�;��M����� ����� � ¥\C��B47:F6�A��D�M�����FìB� ��� � � � � fØ f � =BIO4M6 i GSCP:>AE4M¥74MIn > 1

¢0254\Ap(n)

;34M:<=�AD4�A+G34N2K6MID¥74M8EAOH<I�CKLF4Nk56 i AD=BIR=>knfÙ 4�AD4MIDLNCP:<4a6M2@23AEIDCPH32K4M8

x¢y¢z=>k�;>CK8EADCK: i A�Hr=B8+C@ADC �B4RCK:>AD4�¥747I�8$8*67ADCP8*k@T>CK:<¥

ßPC�àx¢y¢z6�ID4RCK:N6MIDC�A+GBL�4\ADC i H<ID=3¥�ID47898+C5=B:y¢B6M:<;

ßKC@C�àp(xyz) ≤ 3

f��� å(¯Mã�� � � ª ç�ã\äEä*ã�¬ � äEèBé �\ê�ã_çKè�� ° 7çPé � èBé3³/¸� � ê�ê�ã ����ä MèyâEãOµ

¡M4�Ax¢y¢y6�:<;

zJ<4N;BCP8+ADCP: i AXH<=B8*C�ADC��B4�CK:>AD4�¥747I�8R896�ADCK8+k@TBCP:<¥�ßPC�à�6M:<;VßPCPC à f �S4;<4M;B? i 4Rk5ID=BL°ßKC@C�àyA+GB6�A

x = 2a × 3b ¢ y = 2c × 3d ¢>6M:<; z = 2e × 3f � G347ID4 a¢b¢

c¢d¢e¢B6�:<;

f6�ID4�:<=B:<�*:<4M¥�67ADC �B4�CP:>AE4M¥74MI�8 f �QC�A+G3=>?BA�2K=B898�=>k�¥747:<4MI�672PC�A�T ¢�� 4�LN6�T6�8*8*?<LF4RA+GB6�A

x < y < zf�� G347:�ßPC�à0CK8$4M�B?3C �>6M2K4M:>A�AE=

x + z = 2yf ß � à

¡M4�Aδ = gcd(x, y, z)

f�� =�AD4RA+GB6�A (x

δ,

y

δ,

z

δ

) 8*67ADCP8*�r4M8�ß � à�6M:<;

p(

x

δ× y

δ× z

δ

)

≤ p(xyz) ≤ 3f

� G>?<8\¢ (x

δ,

y

δ,

z

δ

) CP8"6S8-=>2@?BADC5=B: f ��=BID4�=7�B4MI9¢(mx, my, mz)

CP8"6S8-=>2@?BADC5=B:�ku=BI4M6 i G

m = 2p × 3q ¢�� G34MID4 p6M:<;

q6MID4 :<=B:<�+:<4�¥�6�ADC��B4aCP:>AE4M¥74MI�8 f�� G347ID47ku=BID4>¢>� 4LF6\TN6M898+?<L�4OA+GB67A

δ = 1f�� G347:

ace = 0 = bdff

� ID=BL¿ß � à+¢�� 4a;<4M;B? i 4�A+GB6�Ax6M:<;

zGB6\�B4RA+G34�896ML�4�H<6MIDC�A�T f

trôMö�ï � fx6M:<;

z6MID4 =3;3; f� G347:Qß � à�ID4�;>? i 478�AD=

3b + 3f = 2c+1 × 3d ¢���C�A+G f > bß58+CK: i 4

x < zà f � k

b > 0¢7A+G347:�A+G34$254�kPA08*CK;<4�=>k�A+G>CK8�4��>?<67ADCK=B:aCP8�;>C��>CK8+C5J>254�JMT

3¢\� G>C i GRku=BI i 4M8

d > 0f

��� �

� G347:δ ≥ 3

¢0� G>C i G i =B:>AEI�6�;BC i AD8"=>?<I�6M898+?<LFH>ADC5=B:VA+GB6�Aδ = 1

f�� G>?<8b = 0

¢k5ID=BL � G>C i GF� 4F;34�;>? i 4"A+GB67Ad = 0

f�� AXku=>2@25=��a8�A+GB67AOß � à�LF6\TVJ<4NID4_�aIDC�A�AE47:V6M82c+1 − 3f = 1

f �<?BA�C@A�CK8�� 4�2P2K� ¤ :<=��a:FßK8-4M4 � ��� à0A+GB6�A�A+G34 =B:B2 T�CP:>AE4M¥74MI�8-=>2@?BADC5=B:=>k2m − 3n = 1

CP8(m, n) = (2, 1)

f�� G>CK8�2K4M6�;38zAE=(x, y, z) = (1, 2, 3)

ftrôMö�ï�� f

x6M:<;

z6MID4 4-�B47: f� G347:

yL�?<8+A�J34�=3;3;Q6M:<;

d ≥ 1ßu=�A+G347Iæ��CP8-4

1 = y > xà f � ID=BL¶ß � à+¢3� 4;<4M;B? i 4�A+GB6�A

b = f = 0fO� G>?<8\¢yß � àzID4�;>? i 478�AE=

2a + 2e = 2 × 3df � =�AD4 A+GB6�A

z > y ≥ 3¢389=�A+GB6�A

e ≥ 2f�� CK: i 4RA+G34aI�C5¥7G�Az8+C5;34 =>kyß � à�CK8$;>C��>CK8+C5J>254aJMT

2JB?BAz:<=�AJMT

4¢�A+G>CP8�ku=BI i 478

a = 1f�� G347:�ß � à�CP8

3d − 2e−1 = 1f �<?BA�C@A�CK8�� 4�2P2 ¤ :<=��a:Fß5894�4

� � � à�A+GB67A�A+G34�=B:B2�T�CP:>AE4M¥74MI$89=>2P?BADCK=B:38R=>k3m − 2n = 1

6MID4(m, n) = (1, 1)

=BI(m, n) = (2, 3)

f�� G>CK8�2K4M6�;38zAE=(x, y, z) = (2, 3, 4)

=BI(x, y, z) = (2, 9, 16)

fg�i+i =BID;BCP:<¥zAE=XA+G34�CK:BC�ADCK672�ID47LF6MI ¤ ¢�A+G34�;<478*CPID4�;OAEI�CKHB25478�6MID4�A+G3=B894z� G>C i GaGB6\�B4=B:<4 =>krA+G34Rku=>2P2K=���CK:<¥Xku=BIELN8$ßu=BI�A+G347CPIzH<4MIDLN?BA�67ADCK=B:38Eà+ì(2p × 3q, 2p+1 × 3q, 2p × 3q+1)

¢(2p+1 × 3q, 2p × 3q+2, 2p+2 × 3q)

¢(2p+1 × 3q, 2p × 3q+2, 2p+4 × 31)

¢� G34MID4

p6�:<;

q6MID4a:<=B:<�+:<4�¥�6�ADC��B4�CP:>AE4M¥74MI�8 f

� ï_óæï �*ï�î �_ï7ö� � � � f � CK4MIDH3CP:38 ¤ C�¢ �\� � �QH3ID=3JB2DÝ4MLF4M8 ;<4NA+G0�4M=BIDCK4w�4�2D�47L�47:>AD67CKID4�;<478 :<=BLFJ3ID478 �M¢H<J f � � � ��� 4\AzHrJ f � � � ��� f¯ f ¡M4\A

ABCJ<4z6zAEIDCP6�:<¥\2K4B¢\6�:<;X2K4�A

DJ<4�6�Hr=>CP:>Ay=B:

AB8+? i G�A+GB6�A

4AD = ABf� G34"GB6M2@ku�+2@CK:<4

`CP8O;BI�6��a:�=B:NA+G34"8*6�LF4�8+C5;34"=>k

AB6�8

C¢<8+A�6�IæADCK:<¥ak5ID=BL

D6M:<;LF6 ¤ CP:<¥F6�:�6�:<¥\2K4Q=>k

�C�A+G

DA¢�� G34MID4

θ = ∠ACBf�� k�A+G34 i CKI i ?<L i CKI i 2K4Q=>k

ABCL�4M4�A�8zA+G34 GB6M2@ku�+2@CK:<4

`6�A

P¢>8-G3=���A+GB6�A

PB = 2PDf

��� å(¯Mã�� ��� ° ç�â �yã-å�¬ Bê�çPå@å�ã ��� � �Bã\è�� ��ä MèyâEã����è�� ��ã ��� ä+ã ���oµ ��Mè � �åPå � � X± �<ã\è���1² ��® � �oµ �"ã���ç$¯Mã`¬ 3ê 7ç@åPå�ã�· é$é � å �BêKç � è1µ

¡M4�AQJ34�A+G34w8-4 i =B:<;nH<=>CK:>AF=>kOCK:>AD4MI�894 i ADCK=B:h=>kXA+G34�2@CK:<4

DP��C�A+G�A+G34i CKI i ?<L i CKI i 2K4 ß5894�4R�r¥\?<ID4\à f�� G347:

∠ADQ = ∠BDP = π − ∠ADP = π − ∠ACB = ∠AQB6�:<;∠QAD = ∠BAQ

f¿� G34MID4�ku=BID4B¢4ADQ

6M:<; 4AQB6�ID4 8+CKL�CP2P6�I f

� G347: AD

AQ=

AQ

AB

f � CP: i 4AB = 4AD

¢C@ASku=>2@25=��a8SA+GB67A

AB = 2AQf �<?BA

4PDB6�:<; 4AQB

6MID4O6M2P8-=a8*CPLNC@2K6MI-¢8*CP: i 4∠BDP = ∠AQB

ßPk5ID=BL 6�J3=M�B4_à6�:<;∠DPB = ∠QPB = ∠QAB

f� ID=BL�A+G>CP8�8*CPLNC@2K6MIDC�A�T 6M:<; A+G34°ID4�2K6��ADC5=B:

AB = 2AQ¢�� 4 i =B: i 2@?r;<4QA+GB6�A

PB = 2PDf

.

..

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

..

..

..

..

..

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...............................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................

.

..

..........

.

.

.

.

.

..

........

.

..

.

..

.

..

..

..

........θ

.

.

.

.

.

.

.

.

.

.

..

..... θ

..

.

..

.

.

..

..

..

.

..

.

..

.

..

..

.

...

..

A

B C

D P

Q

`

�����

á f ¡M4�Af : � → �

J<4�6 H<4MIDLN?BA�67ADCK=B:�=>k<A+G34a894�A�=>k 6M2@2rHr=B8+C@ADC �B4RCK:>AD4�¥747I�8 f

ßK6�à � G3=�� A+GB67A�A+G347ID4"CP8O6�:�6�I�C@A+GBLF4�ADC i H3ID=3¥�ID4M8*8*CK=B:a¢a + d

¢a + 2d

¢<� G347ID4d > 0

¢38+? i G�A+GB67Af(a) < f(a + d) < f(a + 2d)

fßuJ�à���?<8+A A+G347ID4XJ34$6�:�6MIDC�A+GBL�4\ADC i H<ID=3¥�ID47898+C5=B:

a¢a+d

¢. . .¢a+2003d

¢\� G347ID4d > 0

¢38+? i G�A+GB67Af(a) < f(a + d) < · · · < f(a + 2003d) �

� g Hr47IEL�?BAD6�ADC5=B:S=>k�CK8R6N=B:<4��DAE=3�*=B:<4"kK?<: i ADC5=B:F� G3=B8-4"CPLF6�¥74"CP8XA+G34 � G3=>254�=>k

�a` A+GB67AzCK8\¢<6akK?<: i ADC5=B:Fk5ID=BL � AE= � 8*? i G"A+GB67Azku=BI�6M2@2 m ∈ � A+G347ID4aCK8$6 ?<:BCK�B?r4n ∈ � 8*? i G�A+GB67A f(n) = m

f ���� å(¯Mã�� � � ª ç�ã�äEä+ã=¬ � äEèBé �\ê�ã_çKè�� ° �çKé � è3é�³R¸� � ê�ê�ã � ��ä��è â+ã ���è�� ¸�ç � � � � � ª � å � � ±�± ��è>çæê � � � å@å�ã �Bã ���RçKè�ê�ã�ä � O¯Mã�è ���¨¸ ��® � �oµ �"ã��7ç(¯Mã�� � � ��· éo­$ä�ç�ê�ãX³ � ��µ

ß56\à�¡M4�Aa = f−1(1)

6M:<;m = f(a+1)

f�� G347:m ≥ 2

f �BTRA+G34$H3CK¥74�=B:<G3=>2K4H<I�CK: i CPH32K4B¢�A+G34�894��>?r4M: i 4 {f(a+2i)}m−1i=0

i 6M:3:<=�A�J<4�LF=B:<=�AE=B:BC i 672P2 Ta;34 i ID476�8+CK:<¥ f� 47: i 4>¢�A+G34MID4$4*òMCK8EAD8

k��C�A+G

0 ≤ k ≤ m−28*? i GXA+GB67A

f(a+2k) < f(a+2k+1)f¡M4�A

d = 2k f�� G347: f(a) < f(a + d) < f(a + 2d)f

ßuJ�à � = f � :�k56 i A-¢�A+G34MID4�;3=34M8N:<=�A�:<4 i 47898*6�I�CP2 T�4*òMCK8EA"8*? i GÉ6�:�6�I�C@A+GBLF4�ADC iH<ID=3¥�ID47898+C5=B:�=>k�2547:<¥_A+G4f�Ù 4��r:<4

f(n) = 4(3i) − n − 1C@k

3i ≤ n < 3i+1f� G347:

fCP8�;<4 i ID4M6M8*CP:<¥X=B:�A+G34�CP:>AE47I��>6M2

[3i, 3i+1)ku=BI06�:�T�CK:>AD4�¥747I

i ≥ 0f � ?<H3Hr=B894A+GB67A

f(a) < f(a + d) < f(a + 2d) < f(a + 3d)ku=BI 89=BL�4

a¢

d ∈ � f� 47: i 4>¢3j ≤ a < 3j+1 ¢ 3k ≤ a + d < 3k+1 ¢ 3l ≤ a + 2d < 3l+1 ¢�6M:<;

3m ≤ a + 3d < 3m+1 ¢���C@A+G 0 ≤ j < k < l < mf�� G34MID4�ku=BID4B¢

2d = (a + 3d) − (a + d) > 3m − 3k+1 ≥ 3k+2 − 3k+1 = 2(3k+1)f

� G>?<8\¢d > 3k+1 ¢�� G>C i G i =B:>AEI�6�;BC i AD8zA+G34�k56 i A�A+GB67A d < a + d < 3k+1 f

ë f ¡M4\AfJ34 6 kK?<: i ADCK=B:Fk5ID=BL°A+G34a8-4\A�=>k�:<=B:<�*:<4M¥�67ADC �B4aCK:>AD4�¥747I�8$CK:>AD=�C@A�8-4�2Pky8*? i GA+GB67A-¢>ku=BI�6M2@2

n ≥ 0¢

ßPC�à (f(2n + 1)

)2 −(

f(2n))2

= 6f(n) + 1¢B6M:<;

ßKC@C�àf(2n) ≥ f(n)

f� =�� LN6�:�T�:B?<L�J34MI�8�2K4M8*8zA+GB6�:

20036MID4RA+G347ID4�CK:"A+G34RCKLN6�¥74a=>k

f ���� å(¯Mã�� ��� ° � � J±�±�ã�� � 7é9é+çPå �� � êuä 7é � � �Mä � � ��ä��è â+ã � �è�� ¸�ç � � � ��� ª � å � � ±�± ��è>çæê � � � å@å�ã �Bã ���RçKè�ê�ã�ä � O¯Mã�è ���¨¸ ��® � �oµ �"ã��7ç(¯Mã�ê�yãXé � å��3êKç � è ��� � � � ��µ

� CK: i 46f(n)+1

CP8z=3;3;y¢�J<=�A+Gf(2n+1)−f(2n)

6M:<;f(2n+1)+f(2n)LN?<8EA�J<4 =3;3;y¢3JMTNßKC à f�� k

f(2n + 1) − f(2n) ≥ 3¢>A+G347:

f(2n + 1) + f(2n) ≤ 13

(

6f(n) + 1)

= 2f(n) + 136�:<;

2f(2n) =(

f(2n+1)+f(2n))

−(

f(2n+1)−f(2n))

≤ 2f(2n)+ 13

−3¢

��� �

� G>C i GOCP8�6 i =B:>AEI�6�;>C i ADCK=B: f�� G347ID47ku=BID4>¢f(2n+1)−f(2n) = 1

¢76�:<;�ßKC à<ID4�;>? i 4780AD=f(2n+1)+f(2n) = 6f(n)+1

f�� =>2��>CK:<¥�A+G34X8�T38EAE47LS¢�� 4R¥74\Af(2n) = 3f(n)6�:<;

f(2n + 1) = 3f(n) + 1ku=BI�6M2@2

n ≥ 0f� =���� 4�89G3=���¢rJMTFCP:<;B? i ADCK=B:Q=B:

n¢BA+GB67A�C@kyA+G34�J36M8-4 2 ID4MH3ID4M894M:>A�67ADCK=B:Q=>k

nCP8

ak2k + · · · + a12 + a0

¢�A+G34M:f(n) = ak3

k + · · · + a13 + a0f � CK: i 4

f(0) = 0¢<A+G34 i 2P6MCPL CK8�AEID?r4 ku=BI

n = 0f�g 898+?<L�4aA+GB6�A

n ≥ 16�:<;NA+G34 i 2K67CKL CP8AEID?r4�?<H�AD=

n − 1f ��I�C@AD4

n = ak2k + · · · + a12 + a0

f�� G347:f(n) = f(2(ak2k−1 + · · · + a1) + a0)

= 3f(ak2k−1 + · · · + a1) + a0 = ak3k + · · · + a13 + a0

¢� G34MID4XA+G34O2K6M8+A�8EAE47H"CK8�JMT�A+G34OCP:<;B? i ADCK=B:NG�T3H<=�A+G34M8+CK8 f�� CK:3672P2 Ty¢�8*CP: i 4

37 > 2003¢C@A�ku=>2P2K=��a80A+GB6�A

36+ · · ·+3+1 = 1093CP8�A+G34$2P6�ID¥7478+A0:B?<L�J34MI�25478980A+GB6�:

2003CP:A+G34aCPLF6�¥74 =>k

ff�� =�A+G34MID4a6�ID4

27 = 128:B?<LFJ<47I�8$2547898zA+GB6�:

2003CP:�A+G34�CKLN6�¥74=>k

ff

� 4+òMA�� 4�AD?<IE:NAD=N8-=>2@?BADC5=B:38�k5ID=BL =>?<I�ID4M6�;<47I�8$AD=NH<ID=3J>2547LF8X=>k A+G34��y67Þ+6 ¤ G� 6�ADC5=B:3672��w6�A+G34MLN67ADC i 672 £ 2�TBLFHBCK6�;��M�������9����� � ¥\C��B47:NCP: �D�M�����FìB� � � � � � � f¯ f ß � µ ° �� ���è �3ã�ê 7å@ç�ã3¯ à g :<¥\25478B6M:<;

C=>k0AEI�CK6M:<¥\254

ABC6�ID4"6 i ?BAE4 f � C5;34

KN=>k�ID4 i A�6�:<¥\2K4

KLMNJ<4�25=B:<¥�8�AE=N894�¥�LF4M:>A

BC¢3H<=>CK:>A�8

L6M:<;

MJ3472K=B:<¥AE=S894�¥�LF4M:>A�8

AB6M:<;

AC¢0ID4M8*Hr4 i ADC��B4�2�T f ¡M4�A

OJ34�A+G34NCP:>AE47I�8-4 i ADC5=B:SHr=>CP:>AR=>kA+G34�;>CK6�¥7=B:36M2P8�=>k

KLMNf ¡M4\A

C1J34�A+G34�CP:>AE47I�8-4 i ADC5=B:NHr=>CP:>A�=>k 2PCP:<4M8

BO6M:<;

MN¢<6M:<;N254\A

B1J<4�A+G34�CK:>AD4MI�894 i ADCK=B:�H<=>CK:>A�=>ky2@CK:<478

CO6M:<;

LKf 1<ID=M�B4�A+GB6�A2PCP:<4M8

AO¢BB1

¢B6M:<;CC1

6�ID4 i =B: i ?<IDID4M:>A f��� å��3êKç � è ��� ��ã ��� ä+ã ���oµ�� �è�� 7å@å � � X± �<ã\è����1² ��® � �oµ

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................

A

B

B1

B2

C

C1

K

L M

N

O

P�

r s t

p

qu ���*

v

u + v

¡M4�A�2@CK:<4BB1

LF4�4\ALN

6�:<;OA

67AP6M:<;

B2¢�ID4M8*Hr4 i ADC��B4�2�T f ¡M4\A

C2ß5:<=�A8-G3=��a:�CP:"A+G34a;BCP6�¥�I�6ML"à�J<4OA+G34�Hr=>CP:>A�=>k CP:>AE47I�8-4 i ADC5=B:N=>k 2@CK:<4

CC16M:<;

OAf �S4?<8-4aA+G34�ku=>2@25=���CP:<¥a:<=�AD6�ADC5=B:�ì

AL = p¢LB = q ` BK = r

¢KN = LM = s

¢NC = t ` LP = u

¢PO = v

f

��� �

� CK: i 4 4ALM ∼ 4ABC¢0� 4QGB6��B4 p

s=

p + q

r + s + t=

q

r + t` A+GB67ARCP8�¢

p

q=

s

r + t

f� =�� � 4w6�H3H32 T'��47:<472P6M?<8\§ � G34�=BID47L AD=�A+G34�AEIDCP6�:<¥\2K4��DAEI�6�:38��B47I�89672OH<67CKI�8

(4LKN ; B1C)6M:<;

(4LKN ; PB)AD=N¥74�A

LO

ON· t

s + t=

LB1

B1K=

u

u + 2v· r + s

r

f��=B:3894��>?r4M:>AD2 Ty¢ u + 2v

u=

r + s

r· s + t

t` A+GB6�A�CK8\¢

v

u=

1

2

(

(r + s)(s + t)

rt− 1

)

=s(r + s + t)

2rt

fg H<HB2�T>CK:<¥���4M:<4�2K67?<8�§ � G34�=BID47L AD=�A+G34aH36MCPI

(4OAL; B2B)¢�� 4a=3J>A�6MCP:

OB2

B2A=

v

u· q

p + q=

s(r + s + t)

2rt· r + t

r + s + t=

s(r + t)

2rt

f�BTF8�T3LNL�4\AEI�Ty¢

C2;BC �BCK;<478

OACP:NA+G34 896ML�4aI�67ADCK= ` A+GB67AzCK8\¢BA+G34 Hr=>CP:>AD8 B26�:<;

C2i =>CP: i C5;34 f�� G>?<8�¢

AO¢BB1

¢B6M:<;CC1

i =B: i ?<Iz6�AB2 (= C2)

fá f ß ®6µ ° � 7é � ã3¯ à � CK:<;�A+G34XLN6\òMCKLN6M2>6M:<;�L�CK:BCPLF672��>672P?r478z=>k>A+G34X8+?<L

a+b+cCPka2 + b2 ≤ c ≤ 1

f��� å(¯Mã�� � � ° ç�â �yã-å�¬ Bê�çPå@å�ã � � � �Bã\è�� ��ä��è â+ã � �è�� ª ç�ã\äEä*ã�¬ � äEè3é��\ê�ã-ç5è � ° 7çPé � èBé3³¸� � ê�ê�ã ����ä MèyâEãOµ �"ã���ç$¯Mã`¬ 3ê 7ç@åPå�ã�· é$é � å �BêKç � è1µ

� G34aLN6\òMCKLN6M2>�>6M2@?r4�CP81 +

√26�:<;�A+G34aL�CK:BCPLF6723�>672P?r4RCK8 −1

2

f� ?<H<H<=B8-4�A+GB6�A

a2+b2 ≤ c ≤ 1¢�6�:<;X2K4�A

r =√

a2 + b2fy� G347:

0 ≤ r ≤ √c6�:<;

a = r cos θ¢b = r sin θ

¢>ku=BI�8-=BLF4�ID4M672 :B?<LFJ<47Iθf� A�ku=>2@25=��a8�A+GB67A

a + b = r(cos θ + sin θ) = r√

2 sin

(

π

4+ θ

) f� CK: i 4 −1 ≤ sin

(

π4

+ θ)

≤ 16M:<;

r ≥ 0¢r� 4NGB6��B4 −r

√2 ≤ a + b ≤ r

√2f� G347:

a + b + c ≤ c + r√

2 ≤ 1 +√

2¢

8*CP: i 4c ≤ 1

6M:<;r ≤ √

c ≤ 1f�g 2K89= ¢

a + b + c ≥ c − r√

2 ≥ c −√

2√

c =

(√c − 1√

2

)2

− 1

2≥ −1

2

f� G>?<8\¢ −1

2≤ a + b + c ≤ 1 +

√2f��=BID4�=7�B4MI9¢ CPk�� 4 A�6 ¤ 4

a = b = 1√2

6�:<;c = 1

¢rA+G347:a2 + b2 ≤ c ≤ 16�:<;

a + b + c = 1 +√

2¢ � G>C@254 ��C@A+G

a = b = −12

6M:<;c = 1

2

¢r� 4F=3J>A�6MCP:a2 + b2 ≤ c ≤ 1

6�:<;a + b + c = −1

2

f�� G34aH3ID=3=>k CK8 i =BLFHB254\AE4 f

��� �

ë f ß ®�µ ° � 7é � ã3¯ à�¡M4�AyAæ� =�894��>?r4M: i 4M8�=>kBID4M672>:B?<LFJ<47I�8 {an} 6�:<; {bn} J34�8*? i GA+GB67Aa0 = b0 = 0

6M:<;�ku=BIz4M6 i G�Hr=B8+C@ADC �B4RCK:>AD4�¥747In¢

an = a2n−1 + 3

6M:<;bn = b2

n−1 + 2n f��=BLNH<6MID4RA+G34a:B?<LFJ<47I�8

a20036M:<;

b2003f

��� å��3êKç � è ��� ª ç�ã�äEä+ã�¬ � äEè3é��\ê�ã-ç5è � ° �çKé � è3é�³R¸� � ê�ê�ã ����ä��è â+ã"µ� =BI

n ≥ 1¢r� 4FGB6\�B4

a2n > 22n f�� G>CP8Oku=>2P2K=��a8�JMTQCK:<;>? i ADC5=B:V=B:

n¢�8*CP: i 4

a1 = 36M:<;

an > a2n−1

ku=BIn ≥ 1

fÉg :<=�A+G34MI�476�8�TwCP:<;B? i ADCK=B:É89G3=��a8�A+GB6�A2n > n + 3

ku=BIn ≥ 3

f�� G>?<8�¢>ku=BIn ≥ 3

¢�� 4 GB6\�B4a2

n > 22n

> 2n+3 f ß � à�S4 i 2K67CKL A+GB67A

2bn < anku=BIX672P2

n ≥ 3f�� G34 i 2P6MCPLÂCP8OAEID?r4�ku=BI

n = 3¢8*CP: i 4

a3 = 1476�:<;

b3 = 72f 1<ID= i 4M4�;>CK:<¥�JMTFCK:<;>? i ADC5=B:y¢B� 4�6�8*8*?<LF4aA+GB67A�A+G34i 2K67CKLÂG3=>2K;38�ku=BI$8-=BLF4�¥\C �B4M:

n ≥ 3faÚ 8+CK:<¥�A+G34"CP:<;B? i ADCK=B:SG�TBHr=�A+G3478*CP8O6�:<;�ß � à+¢� 4 =3J�AD67CK:

2bn+1 = 2(

b2n + 2n+1

)

= 12

(

(2bn)2 + 2n+3)

< 12

(

a2n + a2

n

)

< a2n + 3 = an+1

¢� G>C i GN4M:<;B8�A+G34RCK:<;>? i ADC5=B: f�BT�?<8+CK:<¥�A+G34 i 2P6MCPL 6M:<; i G34 i*¤ CK:<¥�A+G34S�rI�8EA ku4\� i 6�894M8\¢�� 4V;<4M;B? i 4SA+GB6�Abn ≤ an

ku=BIR672P2n ≥ 0

¢���C�A+G�4��>?<6M2@C@A�TVC@kz6�:<;�=B:B2 TVC@kn = 0

f�� :�H36�IæADC i ?32P6�I9¢a2003 > b2003

f

� = i =BLNH32K4�AD4OA+G>CK8�:B?<LFJ<47I-¢�� 4R2K=3= ¤ 6�A�ID4M6�;<47I�8�§�8-=>2@?BADC5=B:38�AE="H3ID=3JB2K4MLN8$=>kA+G34 Ú�¤ I�67CK:BCP6�:��w6�A+G34MLN67ADC i 672 £ 2 T3LNH3CP6�; ������� � =BIDL¿¥\C��B47:NCP: �D�M�����FìB� � � � ����� fØ f � CK:<;�6M2@2�ID476M2k8*? i GNA+GB6�A�A+G34�ku=>2@25=���CP:<¥a8�T38EAE47L¶=>k�4M�B?<6�ADC5=B:38OGB6�8X6�?<:BCK�B?r48-=>2@?BADC5=B:�ì

x2 + y2 = 2k2 ¢kx − y = 2k

f��� å(¯Mã�� � � ° ç�â �yã-å�¬ Bê�çPå@å�ã � � � �Bã\è�� ��ä��è â+ã � �è�� ª ç�ã\äEä*ã�¬ � äEè3é��\ê�ã-ç5è � ° 7çPé � èBé3³¸� � ê�ê�ã ����ä MèyâEãOµ �"ã���ç$¯Mã`¬ 3ê 7ç@åPå�ã�· é~­$ä�ç�ê�ãX³ � ��µ

� G34O8�T38EAE47L GB6�8z6O?<:BC5�>?r4R8-=>2@?BADC5=B: CPk<6�:<;"=B:B2�T CPkBA+G34O2@CK:<4`��C�A+G"4��>?<67ADCK=B:

kx − y = 2kCK8�A�6�:<¥747:>A�AE=NA+G34 i CKI i 2K4

�C@A+GQ4M�B?<6�ADC5=B:

x2 + y2 = 2k2 fR� G>CK8i =B:<;>C@ADCK=B: i 6M:aJ<4�8+A�67AD4�;�4��>?3C��>672547:>AD2�TR6M80ku=>2P2K=��a87ìBA+G34�I�6�;>CP?<8 √2|k| =>k γ CP8�4��>?<6M2AE=OA+G34X;>CK8EAD6M: i 4$k5ID=BL

`AE=OA+G34�=BIDCK¥\CK:y¢\� G>C i GRCP8 |k · 0 − 0 − 2k|√

1 + k2

f�� G>?<8\¢�� 4�=3J>A�6MCP:A+G34�ku=>2P2K=���CK:<¥ i =B:<;BC�ADC5=B:F=B:

kì2k2 =

4k2

1 + k2

f� =>2��>CK:<¥Oku=BI

k¢�� 4R�r:<;�A+GB67A

k ∈ {0¢1¢ −1} f

���M�

¯ f 1<ID=7�B4XA+GB6�A�ku=BI�6M:�T AEIDCP6�:<¥\2K4B¢�CPkS;<47:<=�AE478zC�AD8z6�ID476�6M:<;

r;34M:<=�AD4M8�A+G34RI�6�;>CP?<8=>k C@A�8�CK:38 i IDCKJ<4M; i CPI i 254>¢�A+G34M:

S

r2≥ 3

√3f

��� å(¯Mã�� ��� ° ç�â �yã-å�¬ 3ê 7ç@åPå�ã � � � �>ã�è � ��ä MèyâEã�� ª ç�ã\äEä*ã ¬ � äEèBé �\ê�ã_çKè�� ° 7çPé � èBé3³¸� � ê�ê�ã � ��ä��è â+ã � Mè �-_rã�� ��åG´�µ ° ��ä-ê ��� � � ¯Mã\ä � ª ����® � �oµ �"ã ��ç$¯Mã ° ��ä-ê ��· éé � å��3êKç � è���± � ��ç ��ã � ���Nê� ã ã ��çæê � ä)µ

Ù 4M:<=�AD4FA+G34Q8+C5;34F2547:<¥_A+GB8"=>k�A+G34FAEI�CK6M:<¥\254QJMTa¢b¢�6�:<;

c6�:<;SA+G34Q8-47LNCK�Hr47IDCPL�4\AE47IaJMT

sfQg H3H32 TBCP:<¥�A+G34 g � 4�� � :<4M�B?<672PC�A�TSAE=�A+G34�Hr=B8+C@ADC �B4F:B?<L�J34MI�8

s − a¢s − b

¢36M:<;s − c

¢�� 4a=3J�AD67CK:3

√

(s − a)(s − b)(s − c) ≤ 13s ` A+GB6�A�CK8\¢

(s − a)(s − b)(s − c) ≤ s3

27

f ß � àÚ 8*CP:<¥ � 4MID=B:y§ 8 � =BIEL�?32K6N6M2K=B:<¥ ��C�A+GVß � à�6�:<;QA+G34 ¤ :<=��a:�ku=BIEL�?32K6

S = rs¢ � 4=3J>A�6MCP:

S2 = s(s − a)(s − b)(s − c) ≤ s4

27≤ S4

27r4

f� G347:y¢ S2

r4≥ 27

¢B6�:<;FG347: i 4>¢ S

r2≥ 3

√3f

ë f ¡M4\AαJ<4F6�ID4M672�:B?<LFJ<47IR8*? i G�A+GB6�AO�>�B4 i =B:3894 i ?BADC �B4�AE47IELN8 =>k�A+G34FCP:B�r:BC�AE48-4M�B?r47: i 4sin α

¢sin 2α

¢sin 3α

¢. . .¢sin nα

¢. . .

6�ID4"I�6�ADC5=B:3672 f 1<ID=M�B4 A+GB67A 7å@åA+G34RAE47IELN8$=>krA+G34a894��>?r4M: i 4a6�ID4�I�67ADCK=B:36M2 f��� å��3êKç � è ��� ª ç�ã�äEä+ã�¬ � äEè3é��\ê�ã-ç5è � ° �çKé � è3é�³R¸� � ê�ê�ã ����ä��è â+ã"µv�ï� � �ô f ¡M4�A

pJ34a6M:�CK:>AD4�¥747Iz8*67ADCP8*k@T>CK:<¥R6M:�TF=>k<A+G34�ku=>2P2K=���CK:<¥Xku=>?<I i =B:<;>C@ADCK=B:387ì

ßPC�àsin(pα) = sin

(

(p + 1)α)

= 0 `ßKC@C�àsin(pα) = sin

(

(p + 2)α)

= 0 `ßKC@CPC àcos(pα) = cos

(

(p + 1)α)

= 0 `ßKC ��àcos(pα) = cos

(

(p + 2)α)

= 0f

� G347:sin α ∈ {−1

¢0¢1} fª ä � ����f ßPC�à � ?<H<H<=B8-4�A+GB6�A

sin(pα) = sin(

(p + 1)α)

= 0fz� G347:

cos(pα) = ±16�:<;0 = sin

(

(p + 1)α)

= sin(pα) cos α + cos(pα) sin α = ± sin αf

� G>?<8\¢sin α = 0

fßKC@C�à � ?<H3Hr=B894 A+GB67A

sin(pα) = sin(

(p + 2)α)

= 0fR� G34M:

cos(pα) = ±16�:<;0 = sin

(

(p + 2)α)

= sin(pα) cos(2α) + cos(pα) sin(2α) = ± sin(2α)f

��� �

� G>?<8\¢sin(2α) = 0

¢�� G>C i G"CPLFHB2PCK4M8zA+GB6�Asin α ∈ {−1

¢0¢1} fßKC@CPC à�6M:<;FßPC���à�6�ID4 i =B:38+C5;34MID4M;�CP:N6 8+CKL�CP2P6�I��a6\T f

� ?<H<H<=B8-4sin(pα)

CK8XI�6�ADC5=B:3672yku=BIp ∈ {n

¢n + 1

¢n + 2

¢n + 3

¢n + 4} ¢ku=BIz89=BL�4RCP:>AE4M¥74MI

n > 0f

trôMö�ï � fR� G34MID4�4*òMCK8EAD8p ∈ {n + 1, n + 2, n + 3} 8+? i GFA+GB6�A sin(pα) 6= 0

6�:<;cos(

(p − 1)α)

6= 0f� G347:

cos α =sin(

(p − 1)α)

+ sin(

(p + 1)α)

2 sin(pα)CK8RI�6�ADC5=B:3672 f � CK: i 4cos(

(p − 1)α) i 6�:VJ34F4+ò�H3ID4M8*8-4M;�6�8R6NHr=>2 T3:<=BL�CK6720CP:

cos α��C@A+GwI�6�ADC5=B:3672 i =34�� i CK4M:>A�8�¢�C�Aaku=>2@25=��a8aA+GB67Acos(

(p − 1)α) CP8"I�67ADCK=B:36M2�ßK6�:<;wCP8:<=B:<�EÞ*47ID= ¢3JMTFG�T3H<=�A+G34M8+CK8Dà f�� G34MID4�ku=BID4B¢

sin α =sin(pα) − sin

(

(p − 1)α)

cos α

cos(

(p − 1)α)

CK8�I�6�ADC5=B:3672 f¡M4�AkJ<4�6�Hr=B8+C@ADC �B4 CP:>AE4M¥74MI f � CK: i 4

sin(kα)i 6�:�J34"4*ò�H<ID4789894�;�6M8X6�Hr=>2 T<�:<=BLNCP6M2<CK:

sin α��C�A+G�I�67ADCK=B:36M2 i =34�� i C547:>AD8\¢>C@A�ku=>2P2K=��a8zA+GB6�A

sin(kα)CK8�I�6�ADC5=B:3672 f

trôMö�ï � f � =BI 476 i Gp ∈ {n+1

¢n+2

¢n+3} ¢9� 4�GB6��B4 sin(pα) cos

(

(p−1)α)

= 0f� G347:y¢0JMTQA+G34�HBC5¥74M=B:<G3=>254�H3IDCP: i CKHB254>¢04�C@A+G347I

sin(pα) = 0ku=BI�Aæ� =N�>6M2@?r4M8=>k

p ∈ {n + 1¢n + 2

¢n + 3} =BI cos((p − 1)α

)

= 0ku=BI�Aæ� ="8*? i Ga�>672P?r478$=>k

pf

� :F47C�A+G34MI i 6�894B¢M� 4RLF6\T"?<8-4OA+G34O2547LFLN6RAE=�;34�;>? i 4OA+GB67Asin α ∈ {−1

¢0¢1} f� :H<6MI�ADC i ?32K6MI-¢

sin αCK8�I�6�ADC5=B:3672�¢>6�:<;"A+G34 i =B: i 2@?<8*CK=B:�ku=>2@25=��a8�6�8�CP:F�y6M8-4 � f

í f�Ù =34M8zA+G347ID4 4*òMCK8EA�6 :B?<LFJ<47Iq ∈ � 6�:<;N6aH<I�CKLF4a:B?<LFJ<47I p ∈ � 8*? i G�A+GB6�A

3p + 7p = 2 · 5q ���� å(¯Mã�� � � ° ç�â �yã-å�¬ Bê�çPå@å�ã � � � �Bã\è�� ��ä��è â+ã � �è�� ª ç�ã\äEä*ã�¬ � äEè3é��\ê�ã-ç5è � ° 7çPé � èBé3³¸� � ê�ê�ã ����ä MèyâEãOµ �"ã���ç$¯Mã`¬ 3ê 7ç@åPå�ã�· é~¯Mã\ä�é+ç � è1µ

� G347ID4�CK8�:<=�8+? i G�H<67CKI(p, q)

fg 898+?<L�4�8+? i G�:B?<L�J34MI�8p6�:<;

q4*òMCK8EA f�� G347:

p 6= 2¢r8*CP: i 4

32 + 72 = 58CK8�:<=�A�=>krA+G34Rku=BIEL2 · 5q

f�� G>?<8�¢pCP8$=3;<; f�� G347:

2 · 5q = 3p + 7p = (3 + 7) · A = 2 · 5 · A¢

� G34MID4A = 3p−1 − 3p−2 · 7 + 3p−3 · 72 − · · · − 3 · 7p−2 + 7p−1 f

� A�ku=>2@25=��a8RA+GB6�AA = 5q−1 ��C@A+G q > 1

ßK8*CP: i 4A > 1

à f � ?<I�A+G347IELF=BID4B¢�8*CP: i 43 ≡ −2 (mod 5)

6�:<;7 ≡ 2 (mod 5)

¢�� 4aGB6��B4A ≡ 2p−1 + 2p−1 + 2p−1 + · · · + 2p−1 ≡ p · 2p−1 (mod 5)

f

��� �

g 8X6�ID478*?32�A-¢p · 2p−1 ≡ 0 (mod 5)

¢ 6M:<;FA+G34�=B:B2 T�Hr=B8*8*CKJBC@2PC�A�TNCK8p = 5

f � CK: i 435 + 75 = 2 · 52 · 341

CP8�:<=�A�=>krA+G34�ku=BIDL2 · 5q ¢�A+G34�H<ID=3=>k CP8 i =BLNH32K4�AD4 f

² f�� CP:<;N672P2<kK?<: i ADCK=B:38f : � → �

8+? i G�A+GB67Af(

xf(x) + f(y))

= x2 + y

ku=BIz672P2x ∈ � 6M:<; y ∈ � f

��� å(¯Mã�� ��� ° ç�â �yã-å�¬ 3ê 7ç@åPå�ã � � � �>ã�è � ��ä MèyâEã�� ª ç�ã\äEä*ã ¬ � äEèBé �\ê�ã_çKè�� ° 7çPé � èBé3³¸� � ê�ê�ã ����ä��è â+ã � �è��G¸�ç � � � � � ª � å � � ±�± ��è>çæê � � � åPå�ã �>ã � �Rç5èyê�ã\ä � O¯Mã�è ���¨¸ �® � �oµ �"ã���ç$¯Mã�� � � ��· é�é � å��3êKç � è�µ

� AaCK8�476�8�T�AD=V�B47IDC@k@T�A+GB6�A�J<=�A+Gf(x) = x

6�:<;f(x) = −x

8*67ADCP8*k@T�A+G34kK?<: i ADC5=B:3672y4��>?<67ADCK=B: f �S4O��C@2P2<H<ID=7�B4RA+GB67A�A+G3478-4�6�ID4OA+G34 =B:B2 T�8-=>2@?BADC5=B:38 f¡M4�AfJ34�6�8-=>2@?BADC5=B: f � 4\A�ADCK:<¥

x = 0CP:�A+G34QkK?<: i ADC5=B:3672�4M�B?<6�ADC5=B:�¥\C �B4M8

f(

f(y))

= yku=BI�6M2@2

y ∈ � f � 4M: i 4B¢>ku=BI�6M2@2 x ¢ y ∈ � ¢x2 + y = f

(

xf(x) + f(y))

= f(

f(x)f(

f(x))

+ f(y))

= f(x)2 + yf

� G>?<8\¢f(x)2 = x2 ku=BIz672P2 x ∈ � f�� k f(1) = 1

¢>A+G347:1 + 2x + x2 = f(1 + x)2 = f

(

1f(1) + f(

f(x))

)2

=(

12 + f(x))2

= 1 + 2f(x) + x2 ¢6�:<;"A+G>?<8

f(x) = xku=BI�6M2@2

x ∈ � f�� CKL�CP2P6�I�2�T ¢>C@k f(1) = −1¢�A+G34M:

1 − 2x + x2 = f(−1 + x)2 = f(

1f(1) + f(

f(x))

)2

=(

12 + f(x))2

= 1 + 2f(x) + x2 ¢6�:<;"A+G>?<8

f(x) = −xku=BI�6M2@2

x ∈ � f

� GB6�A i =BLNH32K4�AD4M8RA+G>CK8 :B?<LFJ<47Ia=>k�A+G34 � � äEè ã�ä f � ��C@2P2�J<4�� =BI ¤ CP:<¥�=B:VA+G34� 4MH>AE47L�J34MI � � äEè ã�ä J<4M¥\CK:3:BCP:<¥�CP:��w6�T ¢�89=w8-47:<;ÉLF4�TB=>?<I�:BC i 4S89=>2P?BADCK=B:38�6M:<;¥74M:<47I�6M2@C@Þ+6�ADC5=B:38�89=3=B: �

��� �

� � � � � ��� �o��� �

�0� � ,���#��X,R%�� ��� ����)*(9,�èyê�ã\äEè BêKç � è�7å ° 3ê� ã�±�BêKç�â�é�² � �MäEè J±�ã�èyê ��� ² � ­$èBé � � � ��� � ��� � ¬ � � ��BT gOf � f� AE=BID=�Þ*G34-� ¢ g � � 13?rJ>2PCP8-G>CP:<¥�¢r�M�������� � � � � � � � � � � � � ����� �M� �9�B¢3H<6MHr47I+J3=>?<:<;y¢r� ��� H36�¥7478�¢ g�Ú ��� � � f ��� f��ã3¯_ç�ãJ­Oã�� � ��������������� ��� � Mè � Mè � � å@å�ã �Bã �Z_rã\äEè � è ��¬ �

� G34 � :>AD4MID:367ADCK=B:36M2 � =>?<IE:36ML�47:>A"=>k � =��a:38�CK8"6VLN67A+G347LF6�ADC i 6M2�H<ID=3J>2547L��8-=>2 �BCP:<¥ i =BLFH<4�ADC�ADC5=B:�ku=BIQG>C5¥7Gh8 i G3=3=>2�8EAD?r;<47:>AD8�k5ID=BL AD=��a:38QA+GBID=>?r¥7G3=>?BA�A+G34� =BID2K; f�� G34��rI�8EA � =>?<IE:36ML�47:>A =>k � =��a:38aAD=3= ¤ H32P6 i 4�CK: 7 ?<8*8*CP6QCP: ��� �M� �M� � �3¢6�:<;�C@A"GB6M8F¥�ID=��a:�AD=wA+G34VH<=>CK:>A�� G34MID4�AD=��a:38�k5ID=BL 6M2@2$ID4M¥\C5=B:38F=>k�A+G34Q� =BI�25;H<6MI�ADC i CKH367AD4 f�� G34 � =>?<ID:36�LF4M:>ArAD6 ¤ 4M80HB2K6 i 4�476 i G�TB4M6MI 6�:<; i =B:38+CK8EAD80=>k7Aæ� =X8+A�6�¥74787ìg ?BAD?<LF:S6M:<; � H3IDCP:<¥ f � 6 i GQ8+A�6�¥74NGB6M8XAæ� =QH36�H<4MI�8�ì$6�: � £ �a2K4-�B472�¢ 2547898R;>C � i ?32@AJB?BA�2547898$Hr=>CP:>AD8 ` 6M:<;�6M: � g �O2K4-�B472�¢rLF=BID4�;BC � i ?32�A$6M:<;�LF=BID4�H<=>CK:>A�8 f�� G347ID4�6�ID4Aæ� =a�B4MI�8+C5=B:38$=>ky476 i G"H<6MHr47I-¢ �+?<:BCK=BI�6�:<; � 47:BC5=BI f�� G34 � 4M:BCK=BI�H<6MHr47I�CK8zCK:>AD4M:<;34�;ku=BI 8+AD?r;34M:>A�8"k5ID=BL A+G34�2P6�8EA�Aæ� =VTB476�I�8N=>k�G>CK¥7G�8 i G3=3=>2�ßKCP:�� 6�:36�;36B¢�¥�I�6�;34M8 ���6�:<; � �\àE¢�6�:<;SA+G34 �+?<:BCK=BI�H36�H<4MIOCK8aCP:>AE47:<;<4M;�ku=BIR8+AD?r;34M:>A�8 k5ID=BL¾25=�� 47I�¥�I�6�;34M8 f� :<;BC �BCK;B?<672�8 i =BID478R6�ID4FJB6�894�;V=B:QA+G34FJ34M8EAO=>k�A+G34�ku=>?<IXH36�H<4MI�8�8+?rJ3L�C@A�AD4�;�?<8+CK:<¥A+G34aHr=>CP:>AD8�k5ID=BL A+G34OAE=BH�A+GBID4M4aH3ID=3JB2K4MLN8�=B:F4M6 i G�H<6MHr47I-¢B6�:<;N6MID4a8 i 67254M;FJB6�894�;=B:�A+G34a8+AD?r;34M:>A-§ 8X¥�I�6�;<4�254-�B4�2 f�g AE=��a:y§ 8$8 i =BID4�CP8�J36M8-4M;F=B:�A+G34a8 i =BID4M8�k5ID=BL°A+G34J<478+A$H36�H<4MI�8Xk5ID=BL¿A+G34 AE=��a:y¢B� G34MID4aA+G34":B?<L�J34MI�=>k�H36�H<4MI�8X?<894�;NAE=Q;34�AD4MIDLNCP:<4A+G34�8 i =BID4aCP8XJ36M8-4M;Q=B:�A+G34aAD=��a:y§ 8�Hr=BHB?32K6�ADC5=B: f � AD?r;<47:>AD8�� G3=F4*ò i 4�4M;�6 i 4MIæAD67CK:LNCP:BCKL�?<L¿8 i =BID4R6�ID4�6\�a6�ID;34�;�6a;BCPH32K=BLF6aJMT�A+G34 7 ?<8*8*CP6�: g�i 6�;<47L"TN=>k ��i CK4M: i 4M8 f� G>CP8�J3=3= ¤"i =B:38*CP8+A�8$=>k � 894 i ADCK=B:38�¢B4M6 i G i =B:>AD67CK:BCP:<¥OA+G34a47CK¥7G�A�H36�H<4MI�8�k5ID=BL=B:<4 � =>?<IE:36ML�47:>A-¢7ku=BIrA+G34 � =>?<ID:36�LF4M:>A�8 ��� A+GBID=>?r¥7Ga� � ¢ i =M�B47IDCP:<¥�A+G34�TB4M6MI�8 �M��� �AE=������>� f � 6 i Gw8-4 i ADC5=B: i =B:>AD67CK:38"8-=>2@?BADC5=B:38aAD=w672P2�=>kzA+G34SH3ID=3JB2K4MLN8�k5ID=BL A+GB6�A� =>?<ID:36�LF4M:>A f�� G34MID4$CK8�6X8*LF672P2>6ML�=>?<:>A�=>k3;B?<HB2PC i 6�ADC5=B:�=>kBH<ID=3J>2547LF8\¢�8+CK: i 4X6�ku4_�H<ID=3J>2547LF8�6MH<H<4M6MI�=B:�J<=�A+G�A+G34 �+?<:BCK=BIy6M:<; � 4M:BCK=BI<�B4MI�8+C5=B:38�=>k�6�H<6MHr47I f�� G347ID4�6�ID4� � �R;BC �047ID4M:>A�H<ID=3J>2547LF8�6M2�AE=3¥74\A+G34MI0CK:aA+G34RJ3=3= ¤ fyg 2PCP8+A�=>kr�>�O¥74M:<47I�6M23ID4�ku4MID47: i 478CK8�¥\C��B47:a67A A+G34�4M:<;a=>k�A+G34�J<=3= ¤ ` G3=�� 4-�B4MI9¢M:<=�� G34MID4�CP:RA+G34�J3=3= ¤ CK8�A+G>CK8�LF6�AE47IDCP6M2i C@AD4�; f � G34$H<ID=3J>2547LF8�CP:RA+G>CP8�J<=3= ¤ 6MID4$:<=�A�ku=BI A+G34$LF6�A+G34MLN67ADC i 672P2 TRk56MCP:>A06�A�G3476�IæA f� G3=>?r¥7G�472K4MLF4M:>A�6�I�T�ßPCK:�A+GB67A�A+G34-T�;<=a:<=�A�ID4��>?3CKID4 ¤ :<=���2K4�;3¥74O=>k i 672 i ?32@?<8�¢M2@CK:<476�I6M2K¥74�JBI�63¢ =BI�=�A+G34MI�6MID4M6M8X=>k06�;��>6�: i 4�;�LF6�A+G34MLN67ADC i 8Dà+¢3A+G34-TF6M2@2�ID4M�B?3CPID4�6 i 4MIæAD67CK:;<4M¥�ID4�4�=>k�LN67A+G347LF6�ADC i 6M2z89=BHrG>CP8+ADC i 6�ADC5=B: f � �B47:�A+G34�4M6M8*CK4M8EA�H<ID=3J>2547LF8�ID4M�B?3CPID48-=BLF4"CP:<¥74M:B?3C�A�T f � G34�L�=BID4N;>C � i ?32@A�H<ID=3J>2547LF8RID4M�B?3CPID4N6NG>C5¥7GQ2K4-�B472�=>k�CP:38*CK¥7G�A6�:<;�CKLN6�¥\CP:367ADCK=B:RAE=�89=>2��B4 f g 2KLF=B8+A04-�B4MI�TaH36�H<4MI i =B:>A�6MCP:38�6�A02K4M6M8+A0=B:<4X¥74�=BLF4�AEI�TH<ID=3J>2547LS¢M:<=B:<4�=>k�� G>C i GRCK: i 2@?r;<478�6X;BCP6�¥�I�6ML f�£ A+G347I�H<ID=3J>2547LF8 i =7�B4MI�6MID4M6M8�8*? i G6�8R:B?<LFJ<47I�A+G34M=BI�Ty¢�6725¥74MJ3I�6B¢ i =>?<:>ADCP:<¥Nß5JB?BAX:<=�AX8EAD6M:<;36MID; i =BLFJBCP:367AD=BIDC i 8Dà+¢�6M:<;25=3¥\C i�f � =BLF4"H<ID=3J>2547LF8O6MID4�;BC � i ?32�A�AD= i 2K6M898+CPk@T ¢ 6�8$A+G34-T i =BLFJBCP:<4�=B:<4�=BI$L�=BID48+A�6�:<;B6�ID;�H<ID=3J>2547L A�T3H<4M8 f �w6M:�T�ID4��>?3CKID4NA+G34�H3ID=3=>k�=>kz6�ID4M8+?32@ARI�6�A+G34MIXA+GB6M:�68*CPLFHB254 i 672 i ?32P67ADCK=B: f � 4-�B4MI�6720H<ID=3J>2547LF8O6M898+?<L�4 ¤ :<=���2K4�;3¥74�=>k�8+A�6�:<;B6�ID;�¥�6�LF4M88*? i G�6�8 i G347898\¢ i G34 i*¤ 4MI�8\¢36M:<; i 6MID;38 f

� � �

� G34Vku=>2@25=���CP:<¥�A+GBID4�4�H3ID=3JB2K4MLN8F89G3=>?325;�¥\C��B4V8-=BLF4Sku4�4�2�ku=BI A+G34V8+A�TB2K4�=>kH<ID=3J>2547LF8$4M: i =>?<:>AE47ID4�;�CK:"A+G>CP8�J3=3= ¤ ìoM}Mð �+îBô3 Nï�îMõ � �98"�æðBî �@}�� 8�m�ð�õ+ði "î � � �������jv�ï��>ï�~ ��� �g IDCK¥7G�A+�*6M:<¥\254M;�AEI�CK6M:<¥\254aLN6�;34 =>k�H36�H<4MI�CK8�ku=>25;34�;N6725=B:<¥R6 8+AEI�67C5¥7G�A�2PCP:<4a8-=A+GB67ArA+G34z�B47I�AD4*òX6�A A+G34�IDCK¥7G�A�6�:<¥\2K4 i =>CK: i CK;<478���C�A+G�=B:<4$=>k�A+G34$=�A+G34MI<�B4MIæADC i 4M8=>krA+G34OAEIDCP6�:<¥\2K4a6M:<;N6 �B?<6�;3I�CP2P67AD4MI�672rCK8$=3J�AD67CK:<4M; f

ß56\à �SGB6�A�CP8NA+G34�I�67ADCK=�CP:>AE=w� G>C i GÉA+G34w;BCP6�¥7=B:3672K8�=>kXA+G>CP8Q�B?<6�;3I�CP2P67AD4MI�672;>C��>C5;34 476 i GN=�A+G347I � ß��aH<=>CK:>A�8EàßuJ�à � G>CK8R�>?<6�;BIDC@2K6�AE47I�6M2�CK8 i ?BA�6725=B:<¥aC@A�8X25=B:<¥7478+A�;>CK6�¥7=B:36M2 f � CK:<;FA+G34�6MID4M6=>k�A+G34Q8*LF672P2K4M8EARH3CK4 i 4�=>kzH<6MHr47IOA+GB67AOCK8�=3J�AD67CK:<4M;�C@k�A+G34Q6MID4M6�=>k�A+G34=BI�C5¥\CP:36M23AEI�CK6M:<¥\254RCK8

1f ß��aH<=>CK:>A�8Eà

oM}Mð �+îBô3 Nï�îMõ � �:8��7ï�î �P} � 8�m�ð�õ*ði �î �74:4 4����nv�ï��>ï�~ ���>�g L�=B:<¥a6�894�A$=>k2N

i =>CP:38�¢r6M2@2yC5;34M:>ADC i 672yCK:Q6MH<H<4M6MI�6�: i 4B¢2N − 2

6MID4"ID476M26�:<;26MID4�k56 ¤ 4 fyg :�TOAæ� =RID476M2 i =>CK:38�GB6��B4�A+G34$896ML�4�� 47CK¥7G�A f0� G34�k56 ¤ 4 i =>CK:38GB6��B4�A+G34R8*6�LF4�� 4�C5¥7G�A-¢7� G>C i G CK8�;>C���4MID47:>A�k5ID=BL A+G34�� 4�C5¥7G�A�=>kr6RID4M672 i =>CP: f

� =�� i 6M:�=B:<4 ;BC �BCK;<4RA+G34 i =>CK:38�CP:>AE="Aæ� =N¥�ID=>?<H<8�=>k04M�B?<672rAE=�A�6M23� 4�C5¥7G�A�JMT?<8*CP:<¥R6�JB6M2P6�: i 4a6�AzLF=B8+A4ADCKLF4M8\¢>C@k

ß56\àN = 16 ` ß � H<=>CK:>A�8Eà

ßuJ�àN = 11 � ß��aH<=>CK:>A�8Eà

oM}Mð �+îBô3 Nï�îMõ �7458��7ï�î �P} � 8�� 3 � � îi� � � ������mjv�ï��>ï�~ ����g ID=3= ¤ CP80672P2K=�� 4�;$AE=XLF=M�B4�=B:<4 i 472@2�4�C@A+G347IyG3=BIDC�Þ*=B:>AD672P2 TR=BIB�B4MIæADC i 6M2@2�T f g kPAD4MI64L�=7�B4M8�A+G34OID=3= ¤ �BCP8*C�AE4M; 6M2@2 i 4�2P2P8z=>kBA+G34

8×8i G3478989J<=B6MID;�6M:<;�ID4�AD?<ID:<4�;J36 i*¤ AD="A+G34aCP:BC@ADCP6M2 i 472@2 f 1<ID=M�B4�A+GB6�A�A+G34 :B?<LFJ<47I$=>kyL�=7�B4M8�CK:�A+G34O�B4MIæADC i 6M2;BCPID4 i ADCK=B:V6M:<;QA+G34F:B?<LFJ<47IR=>k�L�=7�B4M8RCP:�A+G34FG3=BI�C@Þ*=B:>A�6M2�;BCPID4 i ADCK=B: i 6�:3:<=�AJ<4a4��>?<6M2 f ß � H<=>CK:>A�8Eà

� G34�8-=>2@?BADC5=B:38XCP:QA+G>CP8�J<=3= ¤ 6MID4N472K4�¥�6M:>AX6M:<;�� 4�2P2 i I�6MkPAD4�; f � G34�LF6\÷�=BIDC�A�T� 4MID4aH3ID4MH36�ID4M;�JMT g :<;�TS¡�C@?Q=>krA+G34 Ú :BC��B47I�8*C�A�TF=>k g 2KJ<47I�A�6 f0g 2@2y=>krA+G34a89=>2P?BADCK=B:386�ID4VAD4MI�894B¢���C@A+Gj4+òMAEI�6M:<4�=>?<8�=BIN4�2547L�47:>AD6MI�T�÷5?<8+ADC@� i 67ADCK=B:38N254�kPA�AE=�A+G34�ID4M6�;<47I f�w6�:�TO8-=>2@?BADC5=B:38yID47ku47I<AE=X6MID4M6M8�=>k�LN67A+G347LF6�ADC i 8yA+GB6�A�G>CK¥7GO8 i G3=3=>278+AD?r;34M:>A�8 � =>?325;:<=�A�:<=BIDLF672P2 T�J34�k56MLNC@2PCP6�I3��C�A+G ¢M=BIrAD=OID4M8+?32@A�80k5ID=BL'L�=BID4�k56MLNC@2PCP6�I 6�ID476�8yA+GB67A�G>C5¥7G8 i G3=3=>2�8+AD?r;34M:>A�8O� =>?32K;S:<=�AOGB6��B4F4M: i =>?<:>AE47ID4�; f � =BIX4*òM6MLFHB254>¢08-=BLF4 i =B: i 47HBA�86�:<;SID478*?32�AD8Ok5ID=BL ¥�I�6�H<GQA+G34M=BI�TS6MID4�?<894�;QCK:S894-�B4MI�672�8-=>2@?BADC5=B:38 f � G>CP8�6M89H<4 i AX=>kA+G34R8-=>2@?BADC5=B:38 i =>?325; i 4MIæAD67CK:B2 T"8+ADCPLN?32P67AD4XA+G34OCP:>AE47ID4M8EAE4M;"ID4M6�;<47I�AE=";3472 �B4OCK:>AD="6M:?<:Bk56�L�CP2@CK6MI�AD=BH3C i ¢3JB?BAzLNCK¥7G�A�;BCP8 i =>?<I�6�¥74OA+G34aLF=BID4 i 6�8+?<6M2rID4M6�;<47I f� G>CP8nJ<=3= ¤ � =>?325; J<4'6Ü�>672P?<6�JB2K4'ID4M89=>?<I i 4�ku=BIw6�:�TB=B:<4�CP:>AE47ID4M8EAE4M; CP:LF6�A+G34MLN67ADC i 672�H<ID=3J>2547L��+8-=>2 �BCP:<¥ f �w6�A+G34MLN67ADC i CK6M:38�CK:��B=>2 �B4�;¶CK:ÂH<ID47H<6MIDCP:<¥LF6�A+G34MLN67ADC i 8 i =BLNHr4\ADC@ADCK=B:38�� =>?325;��r:<;"CP:"C@A�CP:389HBCKI�6�ADC5=B:�ku=BI i ID4767ADCP:<¥XA+G34�CKI�=��a:H<ID=3J>2547LF8 f � C5¥7Gj8 i G3=3=>2O=BI"?<:BC �B4MI�8+C@A�TnLN67A+G347LF6�ADC i 8�8+AD?r;34M:>A�8�� =>?325;��<:<;jC�A?<8-4�kK?32�6M8�H3ID4MH36�I�6�ADC5=B:�ku=BI�6�:�T�LF6�A+G34MLN67ADC i 8 i =BLNHr4\ADC@ADCK=B: f � G34 g ?<8EAEI�6M2@CK6M:�w67A+G347LF6�ADC i 8 � I�?<8+A�89G3=>?325;�J34 i =B:<¥�I�67AD?32P67AD4�;�ku=BIRHB?rJB2@C i 67ADCK=B:w=>k�H3ID=3JB2K4MLN8�=>kA+G>CK8$�B?<672PC�A�T�6�:<;�2K4-�B472 f

� ���

²�ä�ç � �Bê�ã�ê � ° Bê�yã�±���ç�âæç �è� ;>C@AD4�;aJMT"�r6�IDI�Ta�rCKH3I�63¢ � I�C ¤aÙRfBÙ 47LF67CK:<4>¢>�w6�IæADCK:�¡ fBÙ 47LF67CK:<4>¢�� � =BL 7 =3;<¥747I�8�¢H3?rJ>2PCP8-G34M;FJMT g �É1B4\AE47I�8�¢ �S4�2P2K4M8+254-T ¢r� g ¢<����� ���� � � � � � � � ��� �9��� � � � ¢<GB6MID; i =M�B47I-¢�òM2PC��z�M�>��H36�¥7478�¢ Ú ��� ��� f ��� f��ã3¯_ç�ãJ­Oã�� � ������ �� ��� ������� �������� � � ���G®�èBç(¯Mã�äæé*ç�ê� ��� ´aãX­ ¬ ä ��èBé�­�ç�â� ���ä+ã��<ã\ä�ç�â9ê � è���´`¬ f

� G34 ° Bê�yã�±���ç�âæç �è CP8S�w6MI�ADCP: 4�6MID;3:<47I-¢$ku=BI � G3=BLA+G34G4�6�A+G34MI�CK:<¥Sku=BI4�6MID;3:<47I�ß&4 � 4�à�CK8z:36ML�4M; f0� G34O�3kPA+G"¥�6�A+G34MI�CK:<¥Oß&4 � 4 � à�CP:N����� � 2K4�;aAD=aA+G>CP8�J<=3= ¤ ¢6�:a4�;>C@AD4�; i =>2@254 i ADC5=B:�=>k�A+G>CKIæA�TR6�IæADC i 2K4M8 i =B:>AEIDCKJB?BAD4�;aJMT�H36�IæADC i CPH<6M:>AD8 f�� G34$6MI�ADC i 254786�ID4�H<ID4 i 4M;<4M;°JMT uè ° ã�± � ä�çJ± ¢Qku=>?<IwH3CK4 i 4M8j;34�;>C i 67AD4�; AE= A+G34�2@C��B478�6M:<;i =B:>AEI�C5J>?BADC5=B:38"=>k � ;��a6�ID; � =BID;<47IE:w6M:<; � =3J>?>TB? ¤ C ��=B89G>C5¥�6�GB6�I�6 f�� G34FADC�AD254Q=>kA+G34 J<=3= ¤ CP8��BA�ADCK:<¥�¢B6�8 � � IDCKJB?BAD4 �O894�A�8�A+G34RAD=B:<4�ku=BI�A+G34O� G3=>2K4 J3=3= ¤ ¢�� G>C i G�:<=�A=B:B2�TRH<ID=7�BCK;<478�CK:>AD472@254 i AD?<6M2�6�L�?<8-47L�47:>A0JB?BA�6M2P8-=R;BCP89HB2K6\T38�A+G34�G>?<LN6�:a8+C5;34�=>k�A+G34LF6�A+G34MLN67ADC i 672 i =BLFL�?<:BC@A�T fg I�ADC i 25478Ük5ID=BL�H<=BH3?32P6�I�6M?BA+G3=BI�8�8*? i G¶6�8 7 6\T3LF=B:<; � LN?32@2�T>6M:y¢Ü1B4�AD4MI�QCK: ¤ 2547I-¢ �947IEI�T � 25= i ?<L�¢76�:<; Ù 47:3:BCK8 � GB6M8-GB6$6�H3Hr476�I�6MLNCK;38EA A+G34$;BC �B4MI�8+C@A�Ty¢_� G>C i GCK8 i GB6M2@2547:<¥\CK:<¥"AD=S8*?<LNLF6MIDC�Þ*4�G34MID4 fNg ku4_� 4+òM6�LNH32K4M8�6�ID4�= ��4MID4M; f � CPI�8+A-¢0k5ID=BL��GBI�CK8O�w6�:B2P6�: ¤ 6B§ 8 �_J3=>?r�B?r4\A�=>k�JBI�6MCP:>AE476�894MI�8 ��ì

g J3=>?r�B?r4\A i =B:>AD67CK:380ID4�;RID=B894M8\¢-� G>C�AE4�ID=B8-478�¢�6M:<;�JB2@?r4�ID=B894M8 f g�iEi =BID;3�CK:<¥RAD=�A+G34�� =BI�CK8EA-¢BA+G34�:B?<LFJ<47I$=>k0ID4�;FID=B8-478�6�:<;�� G>C�AE4 ID=B8-478 i =BL�478AE=100 ` A+G34Q:B?<LFJ<47I =>k�� G>C�AE4QID=B894M8 6�:<;wJB2@?r4�ID=B894M8 i =BLF4M8�AE= 53

f� G34":B?<LFJ<47IX=>k�J>2P?r4"ID=B8-478O6M:<;�ID4M;�ID=B894M8 i =BLF4M8�AD=�2K4M8*8�A+GB6M:�A+GB67A f� =��'LF6M:�TNID=B894M8$=>k�476 i G i =>25=BI�6MID4RA+G347ID4 �� 2PCK;BCP:<¥7� i =>CK:�HB?BÞDÞE25478�¢ 6 i I�TBHBADC iRi ID=B8*8E� =BID; ¢rADC@2PCP:<¥�H3ID=3JB2K4MLN8�¢y6�:<;N�>6�I�C5=>?<8=�A+G34MI i GB672P2K4M:<¥7478�6�H3Hr476�I f�g L�=B:<¥$A+G34 i GB6M2@2547:<¥74M8�6MID4RH<=>2�TB=BL�CK:<=�6 i ADC �BC�ADC5478�¢MCK:<�i 2P?r;>CK:<¥a6M:F?<:B?<8*?<672�6MI�ADC i 254 4M:>ADC�AD254M;V1B=>2�TB=BL�CK:<= � ?<L�J34MI � G34M=BI�TQß � � � àzJMT Ú 2K;BCP8�r6�IEJ36M:38�¢ g :<;BIDCP8 �rC5J>?32PCP8�¢�4�C@25J34MIæA�¡M4�4>¢ g :<;�TÉ¡�CP?�¢06�:<; 7 =3J34MIæA ��67CK:��aI�C5¥7G�A-¢0CP:� G>C i G�A+G34Rku= i ?<8�CP8zA+G34 i =BLFH367ADCKJBC@2PC�A�T�=>kyH<4M:>AD=BLNCP:<=34M8$ßK6�8�CP:NH<4M:>AD=BLNCP:<=34M8zA+GB6�A8-GB6MID4 i =BLFLF=B:OL�?32@ADCPH32K4M8Dà f 1<IDCP: i CKHB254780=>kM:B?<L�J34MI<A+G34M=BI�TO6MID4�CK:>AD4�¥�I�6�AE4M;�A+GBID=>?r¥7G;<4��r:BC�ADC5=B:38X8*? i G�6�8\¢�� g H<=>2�TB=BL�CK:<=

ACP8X8967C5;NAE=Q;>C��>C5;34"6M:<=�A+G34MI$H<=>2�TB=BL�CK:<=

BCPk�6 i =BH�TS=>kBLF6\T�J<4"6�8*8-47L�J>254M;Qk5ID=BL i =BH3CK4M8R=>k

Af � � G34"6�IæADC i 2K4"H<ID=7�BCK;<478R6�r:<4 4+òM6�LNH32K4 =>k0G3=�� LN67A+G347LF6�ADC i 6M23A+G>CK: ¤ CK:<¥�¢B2P6�:<¥\?<6�¥74B¢36M:<;NH32P6�T>kK?32K:<47898$LF4�4\ACK:"A+G>CP8��r472K;F=>k ID4 i ID4767ADCK=B:36M2rLF6�A+G34MLN67ADC i 8 f� =BIDLF6M:Q¡ f � 6�:<;>� 4�25;��aI�C@AD4M8z=B: �_�yG>CK:<478-4O��47I�6�L�C i 13?BÞDÞE2K4�¦�4M8*8-4�2K8 � f,7 =B8*8

�3i*¤ 2547IO89G>CPkPA�8OA+G34Nku= i ?<8OAD=�� =BID;BH32P6�T���C@A+G � g � CK8EAE=BI�TV=>k�A+G34 � 47:<� � �>?<6�ID4 �M¢�6M8G34�4*òM6MLNCP:<4M8OA+G34F;<4-�B4�25=BH3L�47:>A�=>k�A+G3=B8-4N672KH<GB6�J34�ADC i 672�6MIEI�6M:<¥74MLF4M:>A�8�CK:Q� G>C i G6M2@2 i =>2@?<LF:38X6M:<;QID=��a8 i =BIEID4789H<=B:<;FAD= � :<¥\2PCP8-G"� =BID;38 f �<C@2P2�� ?BAD2K4MI9§ 8 � Ù 478*CK¥�:BCK:<¥13?BÞDÞE25478 ��C�A+Gw6���=BLNH3?BAD4MI �Q= ��4MI�8"6�:<=�A+G347I��>C54_� =B:�A+G34VG>CK8EAE=BI�T�6M:<; i ?32@AD?<I�672H32P6 i 4 =>kyHB?BÞDÞE25478 f� G34X?<:B?<8+?<6M2<JB2K4M:<;"=>k i =B:>AEIDCKJB?BADCK=B:38�CP8�G>CK¥7G i 6M2@C5JBID4B¢�6M8�=B:<4O4*ò�Hr4 i AD8�� G34M:¤ 4�47:�LN67A+G347LF6�ADC i CP6�:38�8EAE47HVku=BIæA+G�AD=VG3=B:<=>?<IO89=BL�4M=B:<4"A+G34-TVID4789H<4 i A f 7 4M6�;<47I�8��CP2@2�4M:�÷�=7TXA+G34�LN67AD4MI�CK672�¢�6M8y��C@2P2 i 6M8*?<672�JBID=��a8-47I�8 f�� :<;<4M4�;XA+G34$J3=3= ¤ = ��4MI�8089=BL�4M�A+G>CK:<¥R:<4\� =B:F476 i G �>C54_��CK:<¥ �

� � �

� <?>���D�� �>�B ACB,DGF H3>�@

&F# �r!����a#<#3!$,� ! �X,� �z% �X, � � ���a,

� G34�2K= i*¤ 4MI�H<ID=3J>2547L¿6�H3Hr476�I�8�k5ID4M�B?r47:>AD2�T�CK:FJ<=�A+G�A+G34a894 i =B:<;36MI�TN6�:<;�?<:BC5��B4MI�8+C@A�T i ?<IDIDC i ?32P?<L �D�>¢ ��� f � : � =7�B4MLFJ<47I��M��� � ¢�C�A�6MH<H<4M6MID4�;�=B: � 67ADCK=B:36M2y13?rJ>2PC i7 6�;>C5= § 8 ��Mä8² �å 6�8R6��M13?BÞDÞE2547I �a6�:<;�8-=Q67A�A�6MCP:<4�;S4-�B4M:���C5;34MI i CKI i ?32P67ADCK=B: � ��� f� G34�H<ID=3J>2547LÂ¥7=3478�2@C ¤ 4�A+G>CK8�ì g 8 i G3=3=>2 i =BIDIDCK;<=BI�CP8�2@CK:<4M;N��C�A+G

10002K= i*¤ 4MI�8\¢r6M2@2i 25=B894�; f�� G347ID4S6�ID4

10008+AD?r;34M:>A�8 � G3=w6MID4S8-47:>A LN6�I i G>CP:<¥S;<=��a:�A+G34SGB6M2@2�CP:AD?<IE:S6 i+i =BID;BCP:<¥ AD=FA+G34"ku=>2@25=���CP:<¥�I�?325478 f�� G34"�<I�8+A�8EAD?r;<47:>AO=BH<4M:38R4-�B4MI�TQ2K= i*¤ 4MI f� G34N8-4 i =B:<;S8EAD?r;<47:>A i 2K=B8-478a4-�B47I�TS8-4 i =B:<;�2K= i*¤ 4MI9¢�J<4M¥\CK:3:BCP:<¥ ��C@A+G�A+G34N8-4 i =B:<; f� G34XA+G>CKID;"8+AD?r;34M:>A i GB6M:<¥74M8�A+G34�8EAD6�AE4R=>ky4-�B47I�T�A+G>CPID;"2K= i*¤ 4MI9¢BJ<4M¥\CK:3:BCP:<¥���C�A+G A+G34A+G>CKID; ¢ i 2K=B8*CP:<¥�C@A0C@k3C�A0CK8�=BHr47:�6M:<;�=BHr47:BCK:<¥$C�A0CPkBC�A0CK8 i 25=B894�; f0� G>CK8 i =B:>ADCK:B?r478�¢���C�A+GA+G34

n��� 8+AD?r;34M:>A i GB6M:<¥\CK:<¥RA+G34�8EAD6�AE4�=>k04-�B4MI�T

n��� 25= i*¤ 47I-¢3?<:>ADC@2y6M2@2rA+G34 8+AD?r;34M:>A�8GB6��B4��a6M2 ¤ 4�;NA+G34"GB672P2 �a6�T f$� G34�H<ID=3J>2547L CK8�ì �SG>C i GF25= i*¤ 47I�8XID47LF67CK:Q=BHr47:Q6MkPAD4MI6M2@23A+G34a8+AD?r;34M:>A�8�GB6\�B4aLN6�I i G34M; �� G34w6M:38E� 47I�CK8�� 472@2 ¤ :<=��a:�ì � G34�2K= i*¤ 4MI�8�� G3=B8-4�:B?<L�J34MI�8Q6�ID4wH<4MI�ku4 i A8-�>?<6�ID478 ID4MLN6MCP:w=BH<4M:y¢�6�8 =B:B2�T�A+G34�8-�>?<6�ID478�GB6��B4�6M:w=3;3;�:B?<L�J34MIa=>k�;>C��>CK89=BI�8

� ��� f �S4�:<=�AE4�A+GB6�A�A+G>CK8�CP8 AEI�?r4�� G34\A+G34MIRA+G34 i =BIEI�C5;3=BI i =B:>A�6MCP:381000

=BIa6M:�T=�A+G34MI�:B?<L�J34MI�=>k 25= i*¤ 47I�8 f� :�A+G>CP8�:<=�AE4�� 4VH3ID4M894M:>A�8-=BLF4S8*CPLFHB254QAD4 i GB:BCK�B?r478�ku=BI";34M672PCP:<¥F��C@A+G�6M:4*ò7AE47:38*CK=B:�=>k�A+G>CP8�H<ID=3J>2547L f�g A�A+G34F=>?BAD894�A-¢ � 4���CP8-G�AE=S4*ò7AE47:<;�=>?<I$A+GB6�: ¤ 8RAD=

�9=34N�<?rG>2K4MI�ku=BI�2K4M:<;>CK:<¥�G>CK8�6�A�AE47:>ADC5=B:N6M:<;N8-GB6MIDCP:<¥�G>CK8�CK:38+C5¥7G�A�8 f� k�� 4$6�¥�ID4M4�A+GB67AyA+G34�8EAD?r;<47:>AD8�6MID4�:B?<L�J34MID4M; 6M:<;�A+GB6�A-¢\� G34M:a8-47:>A0LF6MI i G3�CK:<¥�¢�8+AD?r;34M:>A

n��CP2@2 i GB6M:<¥74XA+G34R8EAD6�AE4R=>k 4-�B47I�T

n��� 2K= i*¤ 4MI�J34�¥\CP:3:BCK:<¥$��C�A+G 2K= i*¤ 4MI

n¢<A+G347:�C@A�CP8 ¤ :<=��a:FA+GB67A�� 4 i 6�:�2K4M6\�B4 Mè � i =>2@254 i ADC5=B:Q=>k�2K= i*¤ 4MI�8O=BH<4M:QJMT�;>CK89�H<6�A i G>CP:<¥OH3ID4 i CP8-4�2�TaA+G34OIDCK¥7G�A�8*?rJB8-4\A�=>k<8+AD?r;34M:>A�8 � � � f�� G>CK8�2K4M6�;38�AD=�89=BL�4XCP:>AE47I+�4M8EADCK:<¥OH<ID=3J>2547LF8 f�� =BI�CP:38+A�6�: i 4B¢M� 4 ¤ :<=��ÜA+GB67A�8-47:<;BCP:<¥R6M2@23A+G34�8EAD?r;<47:>AD8�25476��B478A+G34S89�B?<6MID4�2K= i*¤ 4MI�8�=BH<4M: f �SG>C i Gw8*?rJB8-4\A�=>k�8+AD?r;34M:>A�8"LN?<8EA�J<4�894M:>ARAE=�25476��B4H<ID4 i CK89472 T�A+G34 i ?rJ<4�2K= i*¤ 4MI�8�=BH<4M: � � =��¶6�J3=>?BA�A+G34�ku=>?<IæA+G�H<=�� 4MI�8 � �S4F��CP2@28-G3=���8-G3=BIæAD2�TRA+GB6�A�A+G34MID4$CK8�6X8+CKLNH32K4�8-=>2@?BADC5=B:RAE=OA+G34M894X�B?r478+ADCK=B:38 f � :aA+G34�L�476�:<�ADCKLF4B¢�� 4X8+A�67AD4XCP: kK?32P23¥747:<4MI�672PC�A�TaA+G34 ã <ê�ã\è �<ã ��å � â�3ã\ä �yä � �Må�ã�± ì�4�C �B4M:�6O8*?rJB8-4\A=>kzA+G34�25= i*¤ 47I�8�¢�� G>C i G�8EAD?r;<47:>AD8�8-G3=>?32K;�J34S;>CK8*H<6�A i G34M;�AD= ¤ 4M4MH�A+G3=B8-4�2K= i*¤ 4MI�8=BHr47: � ��=B:��B47I�8-4�2�T ¢ ¥\C �B4M:Q6�8*?rJB8-4\A�=>kyA+G34�8EAD?r;<47:>AD8\¢B� G>C i GN25= i*¤ 47I�8���C@2P2�J34 254�kPA=BHr47:F67kPAE47I�A+G34-T�LF6MI i G ��S4N:<=�AE4"A+GB67A�A+G347ID4F6MID4F894-�B4MI�672�H<ID=3J>2547LF8RCK:>AD4MI�8*Hr47I�8-4M;�A+GBID=>?r¥7G3=>?BAXA+G34ID4MLN6MCP:<;<47I�=>k�A+G>CK8�;BCP8 i =>?<I�8-4 f0� G34�CKLNH<6�ADC547:>A0ID476�;34MI�LN6�TX��CP8-GXAD=�6�A�AE47LFH>AyA+G3478-4��C@A+G3=>?BAXID476�;>CK:<¥ A+G34NLF=BID4N¥74M:<47I�6M2�ID478*?32�AD8XA+GB6�A��<2@2�A+G34N8*H<6 i 4NJ34�Aæ� 4M4M:QA+G34ML f

�SG>CP2K4F8+? i+i 4M8*8RCK8 i 4MIæAD67CK:B2 TVH<=B898+C5J>254>¢yA+G34�¥747:<4MI�672�ID478*?32�AD8�H<ID=7�BCK;<4N6�LF4M6M:38�ku=BIAD6 i*¤ 2@CK:<¥RLF=B8+Az=>krA+G34a8*Hr4 i CP� i �>?r4M8EADC5=B:38���C@A+GN¥�ID4767AD4MIz4�� i C547: i T f� �*}O��~Pï� � f � G3=���A+GB6�A�A+G347ID4 i 6M:FJ<4a:<=�Aæ� =�;BCP8+ADCP: i Az8-4\AD8$=>ky8EAD?r;<47:>AD8�� G3= ��CP2@225476��B4�=BH<4M:XA+G34�8*6�LF4�8-4\A�=>kM25= i*¤ 47I�8 f�� CK:>A_ì�4�C �B4M:XAæ� =X894�A�8�=>k�8+AD?r;34M:>A�8�¢ i =B:38+C5;34MIA+G34�25= i*¤ 47I�� G3=B8-4�:B?<LFJ<47IzCP8zA+G34a8*LF672P2K4M8EA�� G34MID4RA+G34�8-4\AD8$;>C���4MI f

� �"!$#&%(' )+*+,c© -/."."0 �21/3&1$4"' 1/365718,8*"9+:;18,<' =>1/?2@/�&=<' 9<,(#

� ���

7 4�AD?<ID:BCK:<¥"AD=V=>?<IOLN6MCP:�AE=BHBC i ¢�C@A�CK8�8-G3=��a:�CK: � � � A+GB67AR4�C@A+G347I$�B47I�8*CK=B:�=>kA+G34�4*ò7AE47:<;<4M;wH3ID=3JB2K4ML 6ML�=>?<:>A�8aAE=�89=>2��>CK:<¥F6M:m × m

:<=B:<�*8+CK:<¥\?32P6�I�8�TB8+AD4ML=>ky2@CK:<476�I�4M�B?<6�ADC5=B:38$LF=3;B?32K=2¢�� G347ID4

mCP8�A+G34RAD=�AD672 :B?<L�J34MI�=>ky2K= i*¤ 4MI�8�CK:�A+G34i =BIDIDCK;<=BI f �SG>C@254zA+G34$8�TB8+AD4ML GB6M8�6�?<:BC5�>?r4�89=>2P?BADCK=B:Rku=BIy6�:�TR8*?rJB8-4\A0=>k�A+G34�2K= i*¤ 4MI�8ßu=BI06�:�Ta8+?rJ3894�A�=>k�A+G34�8+AD?r;34M:>A�8EàE¢M�r:<;>CK:<¥�C�A�A+G>CP80�a6�T�CP8�AD4�;>C5=>?<8\¢7AD=a8*6�TRA+G34$2K4M6M8+A f

� 47: i 4�ku=BI�A+G ¢y� 4"��CP2@2�6�8*8*?<LF4�A+GB67ARJ<=�A+GQA+G34F:B?<L�J34MI�=>k�25= i*¤ 47I�8a6M:<;SA+G34:B?<L�J34MIa=>kz8+AD?r;34M:>A�8 6�ID4FCP:B�r:BC�AE4 fS� G34�ID4M8+?32@A�8aCP: � G34M=BID4MLN8 � ¢��B¢�6�:<; � G3=>25;ku=BI�A+G34 i 6�894F=>k�6M:�TS�<:BC@AD4N:B?<L�J34MIm=>k�2K= i*¤ 4MI�8�6M:<;V8EAD?r;<47:>AD8 ` 8+CKLNH32 T�C5¥�:<=BID46�:�TS:B?<LFJ<47I�8 4+ò i 4M4�;>CK:<¥

mCP:S6�:�TV=>k�A+G34N894�A�8X� 4N;BCP8 i ?<8*8 f �S4�6�;3G34MID4"AD=QA+G34i =B:��B47:>ADC5=B:FA+GB67A�A+G34�8-4\A

�=>k�:36�AD?<I�6M20:B?<LFJ<47I�8R;<=3478 è � ê CP: i 2P?r;34

0¢r8-=�A+GB67A�C�Ai =BIDID4M8*Hr=B:<;B8$H<ID4 i CK89472 T�AE=�A+G34 i =BLNH32K4�AD4a8EAD?r;<47:>Az6�:<;�2K= i*¤ 4MI�8-4\AD8 f

� =BI i 4MIæAD67CK:"89H<4 i CP6M238*?rJB8-4\AD8�=>k<A+G34O2K= i*¤ 4MI�8\¢�A+G347ID4OCK8z6�LN? i G"LF=BID4�472K4�¥�6M:>A�a6�T"AD=N6MH<H3ID=B6 i G"A+G34aH3ID=3JB2K4ML A+GB6�:�JMTN8-=>2 �BCP:<¥R6a2K6MID¥74�2PCP:<4M6MIz8�T38EAE47L f�� =��a6MID;A+G>CK8�4M:<; ¢>;347�<:<4$A+G34 é+ç �Mè Bê ��ä+ã =>k36X:367AD?<I�672B:B?<L�J34MI�AE=aJ<4$A+G34�8-4\A�=>k36M2@2>Hr=B8+C@ADC �B44*ò�Hr=B:<47:>AD8�6MH<H<4M6MIDCP:<¥XCP:�A+GB6�A�:B?<L�J34MI9§ 8�H<I�CKLF4Ok56 i AE=BI�C@Þ+6�ADC5=B: f � =BI�=>?<I�H3?<IDHr=B894M8C@A���C@2P2�J<4 i =B:��B47:BC547:>A$AE=N�aIDC�AE4�L�47L�J34MI�8�=>k�A+G34�8*CK¥�:367AD?<ID4N=>k�6N:B?<L�J34MI�CP:S6�89�i 47:<;BCP:<¥a=BID;34MI f � =BIz4+òM6�LNH32K4B¢12 = 2231 6�:<;NGB6�8�8+C5¥�:36�AD?<ID4 {1

¢2} ` 15 = 31516�:<;FGB6M8$8+C5¥�:36�AD?<ID4 {1} f�� =�AD4RA+GB6�A�A+G34a8+C5¥�:36�AD?<ID4 =>k 1 CK8zA+G34 47LFH>A�T�8-4\A-¢ ∅ f� A CK8N4-�>C5;34M:>AaA+GB6�AaA+G34S8-�>?<6�ID4�k5ID4�4V:B?<LFJ<47I�8N6�ID4SH3ID4 i CP8-4�2�T�A+G34S:B?<L�J34MI�8� G3=B8-4�8+C5¥�:36�AD?<ID4�CP8 i =B:>A�6MCP:<4�;"CK:"A+G34�8-4\A {1} f�� G34a89�B?<6MID4M8�6�ID4OA+G3=B894�:B?<L�J34MI�8� G3=B8-4�8*CK¥�:367AD?<ID4�CK8 i =B:>A�6MCP:<4�;RCP:RA+G34$894�A {2

¢4¢6¢. . . } ¢76�:<;OA+G34 i ?rJ34M8�6�ID4zA+G3=B8-4:B?<L�J34MI�8$� G3=B8-4�8*CK¥�:367AD?<ID4�CP8 i =B:>A�6MCP:<4�;�CP:FA+G34"8-4\A {3

¢6¢9¢. . . } f � :S¥747:<4MI�672�¢¥\C��B47:É6S8+?rJ3894�A

A=>k�¢�� 4Q254\A

σ(A);34M:<=�AD4QA+G34�8-4\A�=>k�6M2@2�:B?<L�J34MI�8 � G3=B8948*CK¥�:367AD?<ID4�CP8 i =B:>A�6MCP:<4�;aCP:

Af0� G34X894�A�=>k389�B?<6MID47k5ID4M4X:B?<L�J34MI�8�CK8

σ(

{1}) ¢7A+G34�8-4\A=>k�89�B?<6MID4M8�CP8

σ(

{2¢4¢6¢. . . }

) ¢3A+G34�8-4\A$=>k i ?rJ34M8�CP8σ(

{3¢6¢9¢. . . }

) ¢r6�:<;�8-==B: f"£ k i =>?<I�8-4>¢�LN6�:�T�8+?rJ3894�A�8�=>k�6�ID4�:<=�A

σ=>k�6M:�T>A+G>CP:<¥ f � =BIX4+òM6�LNH32K4B¢ A+G34Hr=�� 47I�8�=>k

2ßu=BIz=>ky6M:�TN89H<4 i C@� i H<I�CKLF4\à�6MID4a:<=�Az=>krA+G34�ku=BIDL

σ(A)ku=BIz6M:�T

Af

� ?<H<H<=B8-4�A+G34�894�A0=>k>8+AD?r;34M:>A�8�CK8�=>k�A+G34�ku=BIELσ(A)

ku=BI�89=BL�4$8+?rJ3894�AA=>k�f

� 8�A+G347ID4�6�: 4�254M¥�6�:>A �a6\TOAE= i GB6MI�6 i AD4MI�C@Þ*4�A+G34�894�A0=>k>2K= i*¤ 4MI�80A+GB6�A ��CP2@2>J<4�254�kPA�=BH<4M:JMTQA+G34M894N8EAD?r;<47:>AD8 � � :<;34�4M;�A+G347ID4�CP8 � �S4�:<4�4M;V=B:<4�L�=BID4N;<4��r:BC�ADC5=B:�CP:V=BID;<47IAE="8EAD6�AE4XA+G34�ID478*?32�A f 4�C��B47:N6�894�AA=>k :36�AD?<I�6M2<:B?<L�J34MI�8\¢>254\A

e(A)J<4R6M2@2<:367AD?<I�672:B?<L�J34MI�8OA+GB67AX6�ID4F¥�ID4767AD4MI$A+GB6M:�=BIX4��>?<6M20AD=�6�: ã3¯Mã�è :B?<L�J34MIOßKCP: i 2P?r;>CK:<¥

0à$=>kA+G34 472K4MLF4M:>A�8$=>k

Af�� =BIz4*òM6MLFHB254>¢>CPk

A = {3¢6¢9¢. . . } ¢>A+G347:

e(A) = {1¢2¢6¢7¢8¢12¢13¢14¢18¢19¢20¢. . . }

= {n ∈ � | n ≡ 0¢1¢2 (mod 6)} f

o �BïM} �*ï� � f ¡M4�AA ⊆ �

f � kO8+AD?r;34M:>A�8σ(A)

6�ID4�;BCP89H367A i G34�; ¢$A+G34M:�2K= i*¤ 4MI�8σ(

e(A)) ID47LF67CK:F=BHr47: f

ª ä � ��� � ¡M= i*¤ 47Im��CP2@2�ID47LF67CK:Q=BHr47:�C@k�6M:<;�=B:B2 TFCPk06M:�=3;<;�:B?<LFJ<47I�=>k08+AD?r;34M:>A�8AE=>? i G�C�A ` A+GB6�AaCP8�¢�C@k m

GB6M8"6�:�=3;<;w:B?<LFJ<47I�=>k$;BC �BCP8-=BI�8�6ML�=B:<¥NA+G34�:B?<L�J34MI�8CK:FA+G34�8+AD?r;34M:>A$8-4\A f � ?<H3Hr=B894 A+GB67A�A+G34�8+AD?r;34M:>A�8σ(A)

6�ID4�8-47:>A$LF6MI i G>CK:<¥�¢ 6M:<;8*?<H3Hr=B894NA+GB67AO2K= i*¤ 4MImGB6�8aH<I�CKLF4Fk56 i AE=BI�C@Þ+6�ADC5=B:

m = pα1

1 pα2

2 · · · pαk

k

fw� =�AE4A+GB67A0A+G34O;BC �BCP8-=BI�8�=>kpαi

i

CK:σ(A)

6MID4X6M2@2B:B?<L�J34MI�8�=>k>A+G34Xku=BIDLpγ

i

¢�� G347ID4γ ∈ A

� � �

6�:<;1 ≤ γ ≤ αi

¢<AE=3¥74\A+G34MIz��C�A+GNA+G34":B?<LFJ<47I1 = p0 ß58+CK: i 4 1 ∈ σ(A)

ku=BI�6M2@28-4\AD8Aà f�� =��

m ∈ σ(

e(A)) CPky6M:<;F=B:B2�T"CPk�476 i G

αi ∈ e(A) ` A+GB67A�CP8�¢>C@k�4M6 i G αiCK8�¥�ID4M6�AE47I�A+GB6M:F=BI�4��>?<6M23AD=2ai

LF4MLFJ<47I�8$=>kA¢>ku=BI�89=BL�4

ai ≥ 0f�� krA+G>CP8�CP8zA+G34i 6M8-4>¢3A+G34M:FA+G34�:B?<LFJ<47I�=>k�;BC �BCP8-=BI�8X=>k

m6�LF=B:<¥�A+G34�8+AD?r;34M:>A�8�� G3=���C@2P2yLN6�I i GCK8

(2a1 +1)(2a2 +1) · · · (2an +1)¢�6M:�=3;<; :B?<LFJ<47I f � 47: i 4�2K= i*¤ 4MI

mID47LF67CK:38=BHr47: f � = i =BLFHB254\AE4�A+G34RH3ID=3=>kæ¢>8+?<H<H<=B8-4�A+GB6�A�8-=BLF4

αi /∈ e(A)¢�896\T

αiCP8�¥�ID4M6�AE47IA+GB6�:�=BI$4��>?<6M2 AD=

2ai − 1L�47L�J34MI�8O=>k

Af�� G347:FA+G34":B?<LFJ<47I�=>k�;>C��>CK89=BI�8O=>k

m6�LF=B:<¥�A+G34$8+AD?r;34M:>A�80� G3=�LN6�I i G i =B:>A�6MCP:38�A+G34$k56 i AE=BI2ai −1+1 = 2ai

¢MLN6 ¤ CK:<¥C@A�4-�B47: f�� G>?<8�¢>2K= i*¤ 4MIm��C@2P2 J34 i 2K=B8-4M; f

�S4O:<=�AE4�A+GB67A�A+G34OH<ID=3=>k =>kBA+G>CK8zID478*?32�A�CK8z6R8EAEI�6MCK¥7G�ADku=BIæ�a6MID;�¥74M:<47I�6M2@C@Þ+6�ADC5=B:=>k�A+G34N8EAD6M:<;36MID;Sku=BIDLF672�H3ID=3=>kz=>k�A+G34N8-=>2@?BADC5=B:�AE=�A+G34F=BIDCK¥\CK:3672�2K= i*¤ 4MI�H<ID=3J>2547Lß5894�4>¢0ku=BIR4+òM6�LNH32K4B¢ � � � à fF� G>CK8�ID4M8+?32@AOLF6 ¤ 4M8a4M6M8�TQ� =BI ¤ =>k�894-�B4MI�672�CP:>AE47ID4M8EADCK:<¥H<ID=3J>2547LF8$ß56ML�=B:<¥XA+G34ML A+G34 =BIDCK¥\CK:3672�à f� �*}O��~Pï� � f�� k�6M2@2 A+G34�8+AD?r;34M:>A�8O6MID4"894M:>A$;<=��a:FA+G34"GB672P2�¢3� G>C i GF25= i*¤ 47I�8XID47LF67CK:=BHr47: �� �*}O��~Pï� � f � k�A+G34�8-�>?<6�ID478�6�ID4$8-47:>A�;<=��a:RA+G34XGB672P2�¢\� G>C i GR25= i*¤ 47I�8�ID47LF67CK: =BH<4M: �� �*}O��~Pï� � f � k�A+G34

n��� Hr=�� 47I�8R6�ID4�894M:>AX;<=��a:�A+G34NGB6M2@2�¢<� G>C i G�25= i*¤ 47I�8RID47LF67CK:=BHr47: �

� �*}O��~Pï� � f � k =B:<4���CK89G34M8�AD= ¤ 4�47H�=B:B2�Ta2K= i*¤ 4MI�:B?<L�J34MI1=BH<4M:y¢7� G>C i G 8+AD?r;34M:>A�88-G3=>?32K;FJ34a8-47:>AzLN6�I i G>CP:<¥ �� G34R2P6�8EA�H3ID=3JB2K4ML 2K4M6�;38�AD="6a:36�AD?<I�6M2 �>?r4M8EADC5=B:�ì � 8zA+G347ID4�6�:"CK:��B47I�8-4�=BHr47I+�67ADCK=B:�AE=NA+G34

ekK?<: i ADCK=B: � � A$CK8O:<=�AX;>C � i ?32@A�AD=Q8-4M4 A+GB67A�A+G34MID4�CK8 f � =BI$6�8*?rJB8-4\A

A=>k A+G34a:367AD?<I�672y:B?<LFJ<47I�8�¢B254\A

A + 1J<4RA+G34 894�A {1} ∪ {n + 1 | n ∈ A} ¢36M:<;;<4��r:<4

f(A)AD=�J34�A+G34X8�TBLFLF4�AEI�C i ;>C���4MID47: i 4O=>k

A6�:<;

A+1 ` A+GB6�A�CK8\¢�LF4MLFJ<47I�8=>krA+G34�?<:BCK=B:A ∪ (A + 1)

A+GB6�Az6�ID4�:<=�AzL�47L�J34MI�8$=>k�J<=�A+G f� �*}O��~Pï� � f � G3=��¿A+GB6�A

e6M:<;

f6�ID4FCP:��B4MI�894Q=BHr47I�67ADCK=B:38 f�� GB67AOCK8\¢�8-G3=��°A+GB6�A

e(

f(A))

= f(

e(A))

= Aku=BI�6�:�T�8-4\A

A=>ky:36�AD?<I�6M2 :B?<LFJ<47I�8 f

1<ID=3JB2K4ML���AE=3¥74\A+G34MIB��C@A+G � G34M=BID4ML � 4M8EAD6�JB2@CK89G$A+G34�ku=>2@25=���CP:<¥�ß@� G>C i GXLF6 ¤ 4M81<ID=3JB2K4ML � 6a89:36MH>à+ìo �BïM} �*ï� � f ¡M4\A

A ⊆ �f � kO2K= i*¤ 4MI�8

σ(A)6MID4VAE=�ID4MLN6MCP:�=BHr47:y¢X8+AD?r;34M:>A�8

σ(

f(A)) LN?<8EA�J<4a;BCP89H367A i G34�; f

t }��*}�~�~�ô�� � f0� G34O8-4\A�=>k<LF6MI i G>CK:<¥X8EAD?r;<47:>AD8�CK8�CK:aA+G34OCPLF6�¥74R=>kσCPk<6�:<;"=B:B2�T C@k3A+G348-4\A�=>k 2K= i*¤ 4MI�8�254�kPA�=BH<4M:�CP8�CK:"A+G34�CPLF6�¥74a=>k

σf

� G34M=BID4ML ��LF6 ¤ 4M8�2PCK¥7G�A�� =BI ¤ =>k<A+G34M894aH3ID=3JB2K4MLN87ì� �*}O��~Pï� �� f � k =B:<4$��CK89G34M8�AE= ¤ 4�47H�=B:B2�TaA+G34 i ?rJ<4X25= i*¤ 47I�8�=BH<4M:y¢7� G>C i G 8+AD?r;34M:>A�88-G3=>?32K;FJ34a8-47:>AzLN6�I i G>CP:<¥ �� �*}O��~Pï� �� f � k�=B:<4R��CP8-G3478�AD= ¤ 4M4MHQ=B:B2 T�A+G34

n��� H<=�� 4MI�8�=BHr47:y¢>� G>C i GN8+AD?r;34M:>A�88-G3=>?32K;FJ34a8-47:>AzLN6�I i G>CP:<¥ �

� � �

�S4"8-4M4":<=�� A+GB67A�A+G34�4*ò7AE47:<;<4M;Q2K= i*¤ 4MI$H3ID=3JB2K4ML CK8O4M6M8*C@2�T�89=>2��B4M;�ku=BI$6M:�T8+AD?r;34M:>A0=BIy2K= i*¤ 4MIy8-4\A A+GB67AyCP8�;<4\AE47IEL�CK:<4M; i =BLFHB254\AE4�2�T�JMT�6�8+C5¥�:36�AD?<ID4�� i =B:>A�6MCP:BCK:<¥8-4\AA ` A+GB67A�CK8\¢rku=BI$894�A�8R=>k�8+AD?r;34M:>A�8R=BI�2K= i*¤ 4MI�8R=>k�A+G34�ku=BIDL σ(A)

ku=BI$89=BL�4Af

� =�� 4-�B47I-¢>A+G>CP8�CP8$6�J389=>2P?BAD472 TN:<=FG34�2KHNCP: i 6�894M8�� G34MID4RA+G34a8+AD?r;34M:>A�=BI�2K= i*¤ 4MI�8-4\ACK8�:<=�A�=>k�A+G>CP8�ku=BIEL f�g ;BC �047ID4M:>AXAD4 i GB:BCK�B?r4 i 6M:�=>kPAD4M:�J<4�JBID=>?r¥7G�AOAD=VJ<476�IXku=BI8*? i G i 6M8-478 f1<ID=3JB2K4MLN8 � 6M:<; � H3ID=M�>C5;34N?<8X��C@A+GS6F:<4 i 47898*6�I�T�CK:38+C5¥7G�A fS� =�AE4"A+GB67A�A+G346�:38D� 4MI�AD=n1<ID=3JB2K4ML � i 6�:ÉJ<4S8EAD6�AE4M;�ì � = ¤ 4M4MH�=BH<4M:É=B:B2�Tw2K= i*¤ 4MIa:B?<L�J34MI

1¢8-47:<;�=B:B2 T�A+G34�8-�>?<6�ID4�k5ID4�4X8EAD?r;<47:>AD8 f ¡�C ¤ 4_��CK894B¢�A+G34�6M:38E� 47I�AD=N1<ID=3J>2547L � i 6�:�J<48+A�67AD4�;yì � = ¤ 4M4MH�=BH<4M:F=B:B2 T"A+G34R25= i*¤ 47I�8zA+GB67Az6MID4

n��� H<=�� 4MI�8\¢B894M:<;�6M2@2r8+AD?r;34M:>A�8� G3=B8-4�:B?<L�J34MI CP8�A+G34SH3ID=3;B? i A"=>k$6V89�B?<6MID47k5ID4M4V:B?<LFJ<47I���C�A+G�6M:

n��� H<=�� 4MI f� G34�89�B?<6MID47k5ID4M4":B?<LFJ<47I�8 i 2K4M6MID2 T�HB2K6\T�6 i I�? i CK672�ID=>2K4 f �S4 2K4�A

S;<47:<=�AE4aA+G34 8-4\A=>k�8-�>?<6�ID4�k5ID4�4":B?<L�J34MI�8 ` S = {1

¢2¢3¢5¢6¢7¢10¢11¢13¢14¢. . . } f�g :<;�ku=BI6�:�TN:36�AD?<I�6M2r:B?<L�J34MI

m¢�� 4R254\A

mS = {ms | s ∈ S} f� �*}O��~Pï� � f�� k�=B:<4O��CP8-G3478zAE= ¤ 4�47H�=B:B2�T"2K= i*¤ 4MI

m=BH<4M:y¢B89G3=���A+GB6�A�=B:<4a89G3=>?325;;BCP89H367A i GN8EAD?r;<47:>AD8

mSf�� CP:>A_ì � 4�4�1<ID=3JB2K4ML � f

o �BïM} �*ï� � f ¡M4�AL ⊆ � J<4"A+G34 i =>2@254 i ADC5=B:S=>k�25= i*¤ 47I�8OAE=SJ34 ¤ 4MH>AR=BH<4M: f�� G34M:8+AD?r;34M:>A

n8-G3=>?32K;SJ<4"CK: i 2@?r;<4M;�CK:QA+G34�8-4\AO=>k�LF6MI i G>CK:<¥"8+AD?r;34M:>A�8RC@k�6M:<;V=B:B2�TQC@k

n ∈ lSku=BI�6�: � � � :B?<LFJ<47I�=>kyL�47L�J34MI�8

l ∈ Lf

ª ä � ��� � ¡M4\AL = {l1

¢l2¢. . . } ��C@A+G l1 < l2 < · · · f�� =�AE4aA+GB6�A L

LF6\TQJ34 �r:BC�AE4=BI�CP:B�r:BC�AE4 f�£ :<4O�a6\T"AD= ¤ 4M4MH�4+òM6 i AD2 T"A+G34R25= i*¤ 47I�8�CP:L=BHr47:�CK8�6M8�ku=>2@25=��a8�ì �rI�8EA8-47:<;V8+AD?r;34M:>A�8

l1Sf�g kPAD4MI�A+G34-T�GB6\�B4FLN6�I i G34M;y¢�=B:B2�T�25= i*¤ 47I

l1CP8 =BH<4M: fF� G34M:8-47:<;�8EAD?r;<47:>AD8

l2Sf�� =�AE4 A+GB6�A�:<=B:<4N=>k�A+G34�8EAD?r;<47:>AD8XCK:

l2S��CP2@2yAD=>? i GQ6�:�T�=>kA+G34��<I�8+A

l2 − 12K= i*¤ 4MI�8 f�� CK: i 4

l1 < l2¢B6MkPAD4MI�A+G>CP8�8-4 i =B:<; i 6�;3ID4a=>ky8EAD?r;<47:>AD8$GB6�8LF6MI i G34�; ¢ =B:B2 TN2K= i*¤ 4MI�8

l16M:<;

l2��C@2P2yJ34�=BHr47: f � k�=B:<4O� 4MID4RAE= i =B:>ADCK:B?r4aCP:NA+G34k56�89G>C5=B:y¢�H<ID4 i CK89472 TSA+G34N25= i*¤ 47I�8�CK:L� =>?32K;�J34Q=BH<4M: f�� =��¿2K4�A

n ∈ � fQ� G34M:n ∈ lS

ku=BI�=B:B2�TQ�<:BC@AD472 T�LF6M:�Tl ∈ L

f � ?<H<H<=B8-4nCP8R6M:V4�2547L�47:>AO=>k�H3ID4 i CP8-4�2�T

k=>k�A+G34�8-4\AD8

lS¢rku=BI

l ∈ Lf�� k

kCP8O4-�B4M:y¢<A+G347:QCP:FA+G34"6�J<=7�B4"8 i 47:36�I�C5=F8+AD?r;34M:>A

n��C@2P2�GB6\�B4�LF6MI i G34�;�6�:S4-�B4M:�:B?<L�J34MIX=>k�ADCPL�478 f�� G>CP8�GB6M8XA+G34�8*6�LF4N4 �04 i AX6M88+AD?r;34M:>A

n:<=�A�LN6�I i G>CP:<¥R67A�672P2 f�� k

kCP8�=3;<; ¢�A+G347:N8EAD?r;<47:>A

n��CP2@2rGB6��B4�LN6�I i G34M;6�:N=3;<;�:B?<LFJ<47I�=>k<ADCKLF4M8\¢>6�:<;�A+G>CK8�GB6�8zA+G34�8*6�LF4a4 �04 i A�6�8�8+AD?r;34M:>A

nLF6MI i G>CK:<¥÷5?<8+A�=B: i 4 f

g 8�8-4-�B47I�6M2�Hr4M=BH32K4�H<=>CK:>AD4�; =>?BAyAD=R?<8�¢�=B:<4 i 6M:aku=BIDLN?32P67AD4 � G34�=BID47L � �>?3C@AD4:367AD?<I�672P2 T�CP:"AE47IELN8�=>k����=3JBC@?<8�CP:��B4MI�8+C5=B: f �<?BA�� 4 GB6\�B4 i G3=B894M:NG34MID4RAD=�H3ID4M894M:>A6�AD=�AD672P2 TN472K4MLF4M:>A�6�I�T�6�H3H<ID=B6 i G f �QC�A+G � G34�=BID47L � CP:FGB6�:<; ¢B8-4-�B47I�6M2 =�A+G34MI i 6M8-478=>krA+G34 4+òMAD4M:<;34�;�2K= i*¤ 4MI�H<ID=3J>2547L¿6MID4O��C�A+G>CK:NID476 i G f� �*}O��~Pï� �54 f ¡M4\A

pJ34�H<I�CKLF4 f�� k�=B:<4���CK89G34M8�AE= ¤ 4M4MH�=BH<4M:�=B:B2 TNA+G3=B894 2K= i*¤ 4MI�8� G3=B8-4�:B?<LFJ<47I�8�6MID4QH<=�� 4MI�8 =>k

p¢�� G>C i GV8+AD?r;34M:>A�8�L�?<8+A�J<4Q;>CK8*H<6�A i G34M; � � I�TA+G>CK8$J<=�A+G ��C�A+G

1 = p0 CP: i 2P?r;34�;�CP:"A+G34�2K= i*¤ 4MI�8-4\A-¢B6M:<;���C�A+G"C@A�:<=�A�CK: i 2@?r;<4M; f� �*}O��~Pï� � � f �SG>C i G�8+AD?r;34M:>A�8FLN?<8EANJ34�;BCP89H367A i G34�;�AE= ¤ 4M4MHn=B:B2�TwA+G34�H3IDCPL�425= i*¤ 47I�8�=BH<4M: �

� � �

g 8�6��<:36M2 =3JB8-47I��>67ADCK=B:y¢M� 4�:<=�AE4OA+GB6�A�A+G347ID4 i 6�:NJ<4�:<=":<=B:<�947LFH>A�T�8*?rJB8-4\AA=>krA+G34�8+AD?r;34M:>A�8�A+GB6�A ¤ 4M4MH38$H3ID4 i CP8-4�2�T�A+G34�2K= i*¤ 4MI�8

A=BHr47:�� G34M:"A+G34 GB6M2@2��a6\TCK8�CP:B�r:BC�AE4 f

o �BïM} �*ï� °� f ¡M4\AA ⊆ � J34�6�:�T�:<=B:<�*4MLNHBA�TQ8+?rJ3894�A$=>k08EAD?r;<47:>AD8 f�� G34M:FA+G34 8-4\A=>k 25= i*¤ 47I�8�254�kPA�=BHr47:FJMT"A+G34M894�8+AD?r;34M:>A�8�CK8�:<=�A

Af

ª ä � ��� � ¡M4�AA = {n1

¢n2¢. . . } ��C�A+G n1 < n2 < · · · f �y=B:38*CK;<47I$25= i*¤ 47I 2n1

f� G34a=B:B2�T�H3ID=BHr47I�;>C��>CK89=BIz=>k2n1

CP:ACK8

n1f � k

2n1 ∈ A6�:<;�A+G34a8EAD?r;<47:>AD8�CP:

ALF6MI i G ¢MA+G34M:"=B:B2 T 8+AD?r;34M:>A�8n16�:<;

2n1��CP2@2�AD=>? i Ga25= i*¤ 47I

2n1¢�6�:<; 8-=�C�A���C@2P2BJ<4i 25=B894�; f ��=B:��B47I�8-4�2�T ¢BCPk

2n1 /∈ A¢BA+G347:�=B:B2�TN8+AD?r;34M:>A

n1��C@2P2<AD=>? i G�25= i*¤ 47I

2n1¢6�:<;S89=QC@A���C@2P2�ID47LF67CK:V=BHr47: f�� :�4�C@A+G347I i 6�894B¢ A+G34N8EAD?r;<47:>AO894�AO;3=34M8R:<=�AXLF6�A i GA+G34�25= i*¤ 47I�8-4\A f

� �*}O��~Pï� � � f�� G34�6�J3=M�B4�H3ID=3=>krk56MC@2K8zCP:�6 i =BIEI�C5;3=BI���C@A+G"6R�r:BC�AE4R:B?<L�J34MIz=>kr25= i*¤ �4MI�8 f � G3=�� A+GB67A�A+G347ID4�CP8X6�894�AACK:Q6M:�T��<:BC@AD4 i =BIEI�C5;3=BIz� G347ID4"8EAD?r;<47:>AD8

A��CP2@225476��B4S=BH<4M:�25= i*¤ 47I�8

Af � k�A+G347ID4S6MID4�L�=BID4FA+GB6�:�Aæ� =�2K= i*¤ 4MI�8 CK:�A+G34SGB6M2@2��a6\Ty¢A+G34MID4O��C@2P2 J34aLN6�:�T�8*? i G�894�A�8 f

�S4"2K4M6\�B4�A+G34�ID476�;34MI���C@A+G�=B:<4"�<:36M20H<ID=3J>2547L f � =BI$6�:�T�8+?rJ3894�AA=>k�A+G34:367AD?<I�672 :B?<L�J34MI�8\¢<;34M:<=�AD4 JMT

A2 A+G34�8-4\A {n2 | n ∈ A} f� �*}O��~Pï� � � f�� G34�89=>2P?BADCK=B:�AD= A+G34�=BIDCK¥\CK:3672325= i*¤ 47I�H3ID=3JB2K4ML 8-G3=��a8�A+G347ID4RCP8�6�8-4\AA��C@A+GFA+G34NH3ID=BHr47I�A�TQA+GB6�A�� G347:S8+AD?r;34M:>A�8

ALN6�I i G ¢y2K= i*¤ 4MI�8

A2 ID4MLN6MCP:V=BH<4M:ß@÷5?<8EA0A�6 ¤ 4A = �

à f � CK:<; 6X:<=B:<�*4MLNHBA�T 8-4\AA8-=RA+GB6�Ay� G34M: 8+AD?r;34M:>A�8

A2 LN6�I i G ¢25= i*¤ 47I�8AID47LF67CK:F=BH<4M: f

� ï_óæï �*ï�î �_ï7ö� � � �y6MI � 6M2 ¤ ß�� 4�4 ¤ 2 TNI�6�;BCK=�H<ID=3¥�I�6ML"àE¢ � =M� f ��� ¢r�M��� � ¢

http://cartalk.com/content/puzzler/transcripts/200546/f

�D� � � f �y6�HB2K6M:�6�:<;�� f ��=B8 ¤ =���C@A�Þ_¢ ° 3ê� ã�±�BêKç�â�é ª ä � �Må�ã�± � � å$¯_ç5è � ¢ � IDC@?<LFH<G¡M4M6MIE:BCP:<¥�¢ � 4\� ��=BI ¤ ¢3�������B¢3H f � f� � � 7zf ¡M47CKJ<=���C�A�Þ_¢ � ��I�C@ADCP:<¥F;>CK8 i ID4�AD472 T��M¢0k5ID=BL � f 4 f�7 =B894M:38EAE4�CK:y¢�4�; f ¢ ��çPé�âDä+ã�ê�ã° Bê�yã�±�3êKç�âæéXç5è�ê�yã � â � � � åPé ¢ g LF4MI�C i 6�:��w67A+G347LF6�ADC i 6M2 � = i C54\A�T ¢ Ù � � g � �8-47IDCK4M8\¢�¦�=>2 f � �3¢ �M��� � ¢BH f ��� f� ��� � fM� =BIEID47: i 4>¢ � � ò7AE47:<;BCP:<¥�A+G34$25= i*¤ 47I0H<ID=3J>2547L��M¢ ° 3ê� ã�±�BêKç�â �ç5è � � �Bâ 3êKç � è

�è�� ��ã_é�ã �ä+â � ¢ ��� ì � ¢��*6�: f �M�����3¢BH<H f �>� ��� f�rID? i 4 � =BIDID4M: i 4 � AD6M: ��6�¥7=B:Ù 4MH36�IæAEL�47:>A�=>k��w6�A+G34MLN67ADC i 8 Ù 47H<6MI�AELF4M:>A�=>k��w6�A+G34MLN67ADC i 87 6�:<;3=>2KH<G3�\�w6 i =B:��y=>2P2K4�¥74 �w6 i 6725478+AD4MIz��=>2@254M¥74g 8-G>2P6�:<; ¢�¦ g � � ��� � � A f 1>6M?32�¢ � � � ��� � �Ú �\g Ú �\gbtorrenc@rmc.edu wagon@macalester.edu

� ���

� � � � � � �

¤ WXPJZ-N/Y1YRWdS&P3ZeQ-W�VJY1^XPgf�hi\aWJj)S)kN/_�NgY�l�^�V3Y � N�ViP�_?mNR\aW�l"WdQ-b)N/ViZiVOWXP)Y�hO^�\abdNRV�Qe\�^XP�h"S&P)YZe^�\.ltS-N��)qdp������3qRrox�p.qoydz�z3{X|1§¨ViN�mN)Z&WdQ S-N(?)^Rhi\OkN/Y�S-N,ViP�_?mNR\aWEQeVOl�Q]�OP3N��OP3N6S-N�hi\aWJj)S�kNR_`N

^�mN�Z)mN�Y/WXP�_tQ�Y�Ya^�VJY�YRWdS&P3ZeQ-W�V¨|�X� ^3�OPJN�hi\aWJj)S)kN/_�NcYRN/\&^Ih�PJj)S QRmNIl�^�VJY�S-N/Y�l"N�P/fIS�^�V3�XP3N/Y�W)� � Q-NUS S-N/Y�lOP � ^�VJ^3l�^� ^�V3�dS�^dQ�Y~N�Z��e\&^�V��� ^dQ�Y&�/|1��^�V3Y�S-NgYV�P)_?mN/\aW)Y~�g���R���8N�Z,�w��S$� ^�V��dS�^dQ�YEhi\XmN � mN)l"N/\&^�S-Nt��\&^�V��� ^dQ�Y.�N�Z1l�^�V3Y6S-NgY�ViP�_?mNR\aW)Y,�R���)���oN�Z1 /��S-NE��\&^�V��� ^dQ�Y�hi\XmN � mN)l"N/\&^S$� ^�V��dS�^dQ�Y)|¨��^�V3Y6S�^tYRN � ZeQ-W�V�l"N/YYRWdS&P3ZeQ-W�VJY.�JS-N1hi\aWJj)S)kN/_�N�YRN/\&^�h�P3j)S QdmN�l�^�VJY�S�^6S�^�V��XPJN�l"N�S�^�hi\(QeV � Q hO^dS-N�Y/WdS&PJZeQ-W�V,hi\XmN/Y/NRVOZ�mN�N�|})^c\XmN)l�^ � ZeQ-W�V�Y/WXP � ^dQ-Z&N8\aNR_`NR\ � Q-NR\�¡dN/^�VO¢w£�^�\ ��¤ N/\.\&Q-NR\�N�Z£�^�\wZeQeV¦¥�WdS]l�YRZ-NgQeV���l"NS$��§¨V�Q-b)N/\&YaQ-Z)mN,l"N�£IW�VOZ]\XmNg^dS ��l�� ^Jb)WdQe\�Z]\&^3lOPRQ-ZOS-N/Y1hi\aWJj)S)kN/_�NgY)|

áO¯Oá© f�ª ä � � � é�«ã ��Mä ° ç �6«7å �[¬0ã�è â �_ã ��¬ ä 7é � ¯ � � � �J±��è>ç�ã f� =>C@A

n?<:F4M:>ADCK4MI�Hr=B8+C@ADC@k f� C

α = 1 +1

12(n + 1)

¢BLF=B:>AEID4MI��>?r4

e <

(

(n + 1)2n+1

(n!)2

)1

2n

< eα ¢

áO¯ ë ± f�ª ä � � � é�«ã ��Mä ° ç �6«7å �[¬0ã�è â �_ã ��¬ ä 7é � ¯ � � � �J±��è>ç�ã f� =>C@A

n?<:N:<=BL�JBID4 47:>ADC547IzH<=B8*C�ADCPk f ��=B:>AEID4MI��>?r4⌊√

n +

√

n + 2 3√

n + 1

⌋

=

⌊√

4n + 4 3√

n + 2

⌋ ¢=�Ý? bxc ;��4M8+C5¥�:<4R2K6aH<6MI�ADCK4 47:>ADCDÝ47ID4 ;34 x

fáO¯ ë Ø f�ª ä � � � é�«ã ��MäT_3ç5ä �7ç@å�´�ç�â ��å ��¬ �Bâ �ä+ã\é�ê��� � �J±��è>ç�ã f

� =>C@Aa¢b4\A

cAEID=>CP8�:<=BL�JBID4M8�I��4M472P8�6�IEJBC�AEI�6MCPID4M8�AE4�2K8��>?r4

a2 + b2 + c2 = 9f��=B:>AEID4MI��>?r4

3 · min{a¢b¢c} ≤ 1 + abc

fáO¯ ë ¯ f�ª ä � � � é�«ã ��MäT_3ç5ä �7ç@å�´�ç�â ��å ��¬ �Bâ �ä+ã\é�ê��� � �J±��è>ç�ã f

� =>C@A A ={

z ∈ � ∗ :∣

∣

∣z +1

z

∣

∣

∣ ≤ 2} f�� =>C@A

n ≥ 2?<:�47:>ADC547I f ��=B:>AEID4MI$�>?r4

8*Cαn ∈ A ¢B6M2K=BI�8 α ∈ A fáO¯ ë á f�ª ä � � � é�«ã ��Mä���ã � ä �Bã�²_é+ç5èyêKé+ç � Mé ��² � ã\é*é �å � è>ç �>ã ����ä �ãMâ+ã f

� =>C@AP?<:QHr=>CP:>A�CP:>A7�4MI�C54�?<I�; § ?<:FAEIDCP6�:<¥\2K4 CK89= i Ý472K4

ABC6\�B4 i

AB = ACf

� =>C@AD4\A

E25478OHr=>CP:>AD8R; § CP:>AE47I�8-4 i ADC5=B:�ID4789H<4 i ADC@k58R;<478R;3ID=>C�AE478

BP4�A

CP6��B4 i25478 i7Û=�A7�4M8$=BH3Hr=B8��4M8 fB� ID=>?>�B47I�254R2@C54�?�;<478$H<=>CK:>A�8

P8+C

PD + DC = PE + EBf

� � �

áO¯ ëOë f�ª ä � � � é�«ã ��Mä���ã � ä �Bã�²_é+ç5èyêKé+ç � Mé ��² � ã\é*é �å � è>ç �>ã ����ä �ãMâ+ã f� =>C@A

P?<:QHr=>CP:>A�CP:>A7�4MI�C54�?<I�; § ?<:FAEIDCP6�:<¥\2K4 CK89= i Ý472K4

ABC6\�B4 i

AB = ACf

� =>C@AD4\A

E25478OHr=>CP:>AD8R; § CP:>AE47I�8-4 i ADC5=B:�ID4789H<4 i ADC@k58R;<478R;3ID=>C�AE478

BP4�A

CP6��B4 i25478 i7Û=�A7�4M8$=BH3Hr=B8��4M8 fB� ID=>?>�B47I�254R2@C54�?�;<478$H<=>CK:>A�8

P8+C

BD + DC = BE + ECf

áO¯ ë í f¦ª ä � � � é�«ã����ä ª ��å���ç � � �Må � ä�ç�� ��XêKåMè�êKç�â8®0è>ç$¯Mã\ä�é+çæê ����¬ � â � Bê � è � �¨¸ �® � � f

£ :�8+?<H<H<=B8-4��>?r4V2K4 i 4M:>AEID4�;B? i 4MI i 2K4�;<478N:<47?3kXH<=>CK:>A�8F;y§ ?<:�AEIDCP6�:<¥\2K4V8-4AEID=>?>�B4X8*?<I02K4 i 4MI i 2K4�CP:38 i I�C@A�;>? AEI�CK6M:<¥\254 f ��=B:>AEID4MI��>?r4X8-=B: 6M:>ADCKH<=3;<4X4M8EA0254�Hr=>CP:>A;<4 � 47?r47I+JB6 i G ¢ i § 478+A+�MÝ6��*;BCPID4R254�Hr=>CP:>A�;<4OAD6M:<¥74M: i 4 ;<478�;347?�ò i 47I i 25478 fáO¯ ë ² f�ª ä � � � é�«ã ��Mä ° Mä�çMèG²7ã�êKç(¯ ���¬��� ä�å � � � � �X±�MèBç�ã f

� =>C@Aa¢

b¢

c4\A

d;<478Ü:<=BLFJ3ID478ÜI��4M472P8ÜH<=B8*C�ADCPk58h6MI+J>C@AEI�67CKID478�AD472P8��>?r4

d = min{a¢b¢c¢d} f ��=B:>AEID47I��>?r4

a4 + b4 + c4 + d4 − 4abcd

≥ 4d[

(a − d)3 + (b − d)3 + (c − d)3 − 3(a − d)(b − d)(c − d)] f

áO¯ ë � f ª ä � � � é�«ã���Mä¶ � é�«ã ¸ �7çPé �o«� ��³R¬ MäEä*ã\ä � �8®�èBç(¯Mã�äæé*ç�êJ«ã ª � å�Bê�ãMâ �<è>ç �>ã �<ã��3ê 7å � ��è ã ��¬ Mä*âEã_å � èyã � � é �� ��è ã f

� =>C@Aa1¢a2¢. . .¢an;34M8�:<=BL�JBID4M8�I��4M472P8�¢\AD=>?<8�HB2P?<8�¥�I�6M:<;38��>?r4

1f ��=B:>AEID47I�B?r4

n∑

k=1

(

1 + logak(ak+1)

)2 ≥ 4n¢

=�Ý?an+1 = a1

fáO¯ ë � f8ª ä � � � é�«ã����ä²\çæê � �O¯ � è��ä � � � � ± «�è ã��é7êKç ��� � �X±�MèBç�ã �yã�ê�¬ � � � �è � è>çæê ��� �¬ �Bâ �ä+ã\é�ê��� � �J±��è>ç�ã f

� Ca¢b4�A

c8-=B:>Az;<478$:<=BLFJ3ID478$I��4�4�2K8�Hr=B8+C@ADC@k58�¢>LF=B:>AEID4MIz�B?r4

a2(b + c − a)

b + c+

b2(c + a − b)

c + a+

c2(a + b − c)

a + b≤ ab + bc + ca

2

fáO¯ ë © f8ª ä � � � é�«ã����ä²\çæê � �O¯ � è��ä � � � � ± «�è ã��é7êKç ��� � �X±�MèBç�ã �yã�ê�¬ � � � �è � è>çæê ��� �¬ �Bâ �ä+ã\é�ê��� � �J±��è>ç�ã f

� =>C@Aa¢b4�A

c2K4M8�2K=B:<¥\?r47?<I�8�;34M8 i7Û=�A7�4M8$; § ?<:"AEIDCP6�:<¥\2K4 f ��=B:>AEID47I��B?r4(b + c)2

a2 + bc+

(c + a)2

b2 + ca+

(a + b)2

c2 + ab≥ 6

f

� �M�

áO¯ í ± f�ª ä � � � é�«ã ��Mä ��µ ¶Jµ � ±�ã�ã\è� � � 7å�ê � � ±�±�ã_å � ª ��Mé�³R¬ 7é f� =>C@A

ABC?<:�AEI�CK6M:<¥\254jCK89= i Ý472K4j6��B4 i

AB = AC4�AQAE4�2"�B?r4n2�§ 6M:<¥\254

BAC = 100◦ f � =>C�A D2K4QHr=>CP:>Aa8*?<IR2K4QH<ID=>2K=B:<¥74MLF4M:>A ;34

AB67?r�9;3472DÝ6S;<4

A4�A�AD472 �B?r4AD = BC

fB� ID=>?>�B47I�2�§ 6�:<¥\2K4ADC

ff f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

áO¯Oá© f ª ä � � � é�ã � � � ° ç ��«�å��¬0ã�è â �_ã ��¬ ä�Mé � ¯ ��� � ±��è>ç f¡M4�A

nJ34a6aHr=B8+C@ADC �B4RCK:>AD4�¥747I f�� k

α = 1 +1

12(n + 1)

¢>H<ID=7�B4RA+GB67A

e <

(

(n + 1)2n+1

(n!)2

)1

2n

< eα ¢áO¯ ë ± f ª ä � � � é�ã � � � ° ç ��«�å��¬0ã�è â �_ã ��¬ ä�Mé � ¯ ��� � ±��è>ç f

¡M4�AnJ34a6aHr=B8+C@ADC �B4RCK:>AD4�¥747I f 1<ID=M�B4OA+GB67A⌊√

n +

√

n + 2 3√

n + 1

⌋

=

⌊√

4n + 4 3√

n + 2

⌋ ¢� G34MID4 bxc ;<47:<=�AE478�A+G34RCK:>AD4�¥747IzH<6MI�A�=>k x fáO¯ ë Ø f ª ä � � � é�ã � � �O_3çKä ��çPå�´�ç�â �7å ���¬ �>â � Mä*ã_é7ê� � � ±�MèBç f

¡M4�Aa¢b¢cJ<4R6�:�T"ID4M672r:B?<LFJ<47I�8�8*? i G A+GB6�A

a2 + b2 + c2 = 9f 1<ID=7�B4OA+GB6�A

3 · min{a¢b¢c} ≤ 1 + abc

fáO¯ ë ¯ f ª ä � � � é�ã � � �O_3çKä ��çPå�´�ç�â �7å ���¬ �>â � Mä*ã_é7ê� � � ±�MèBç f

¡M4�A A ={

z ∈ � ∗ :∣

∣

∣z +1

z

∣

∣

∣ ≤ 2} f ¡M4\A

n ≥ 2J34O6�: CP:>AE4M¥74MI f 1<ID=7�B4�A+GB6�A-¢

CPkαn ∈ A ¢�A+G347: α ∈ A fáO¯ ë á f ª ä � � � é�ã � � � �6µ�²_é+ç5èyêKé+ç � Mé �,² �yã_é9é 7å � èBç �ç ����ä+ã�ãMâ+ã f

¡M4�AABC

J<4Q6M:�CP8-=B8 i 4�25478�AEI�CK6M:<¥\254N��C�A+GAB = AC

¢�6M:<;�254\APJ<4Q6M:CK:>AD4MI�C5=BI�H<=>CK:>A f ¡M4�A�A+G34a2@CK:<478

BP6�:<;

CPCK:>AD4MI�894 i A�A+G34�=BH<H<=B8*C�AE4a8*CK;<478�6�AzA+G34Hr=>CP:>AD8

D6M:<;

E¢BID4789H<4 i ADC �B472 T f�� CK:<;�A+G34�2K= i ?<8�=>k

PCPk

PD + DC = PE + EBf

áO¯ ëOë f ª ä � � � é�ã � � � �6µ�²_é+ç5èyêKé+ç � Mé �,² �yã_é9é 7å � èBç �ç ����ä+ã�ãMâ+ã f¡M4�A

ABCJ<4Q6M:�CP8-=B8 i 4�25478�AEI�CK6M:<¥\254N��C�A+G

AB = AC¢�6M:<;�254\A

PJ<4Q6M:CK:>AD4MI�C5=BI�H<=>CK:>A f ¡M4�A�A+G34a2@CK:<478

BP6�:<;

CPCK:>AD4MI�894 i A�A+G34�=BH<H<=B8*C�AE4a8*CK;<478�6�AzA+G34Hr=>CP:>AD8

D6M:<;

E¢BID4789H<4 i ADC �B472 T f�� CK:<;�A+G34�2K= i ?<8�=>k

PCPk

BD + DC = BE + ECf

� � �

áO¯ ë í f ª ä � � � é\ã�� ��� ª ��å ��ç������Må � ä�ç�� ��XêKå �èyêKç�â5®�èBç(¯Mã�äæé*ç�ê� ��¬ � â� � Bê � è � �¨¸ �® � � f

� ?<H<H<=B8-4�A+GB67A0A+G34 i 4M:>AEID4R=>kBA+G34R:BCP:<4��+Hr=>CP:>A i CPI i 254R=>k<6XAEIDCP6�:<¥\2K4X2PCK4M8z=B: A+G34CK: i CPI i 254R=>kBA+G34�AEIDCP6�:<¥\2K4 f � G3=��hA+GB6�A�C�AD8�6�:>ADCPHr=3;B6M2BHr=>CP:>A�CP8�A+G34 � 4�?r4MIEJ36 i G�1B=>CK:>A `A+GB67A�CP8�¢BA+G34 Hr=>CP:>A�� G347ID4aA+G34 :BCK:<4M�*H<=>CK:>A i CPI i 254 6M:<;NA+G34aCK: i CPI i 254 6�ID4�AD6M:<¥74M:>AzAD=4M6 i GF=�A+G347I fáO¯ ë ² f ª ä � � � é�ã � � � ° �ä�ç �è?²Mã�êKç$¯ ��¬��� ä�å � � � � ±��è>ç f

¡M4�Aa¢b¢c¢dJ34"6M:�TQHr=B8+C@ADC �B4�ID4M6720:B?<LFJ<47I�8$��C@A+G

d = min{a¢b¢c¢d} f1<ID=M�B4OA+GB67A

a4 + b4 + c4 + d4 − 4abcd

≥ 4d[

(a − d)3 + (b − d)3 + (c − d)3 − 3(a − d)(b − d)(c − d)] f

áO¯ ë � f�ª ä � � � é\ã�� ���¦¶ � é�«ã ¸ �7çPé �o«� ��³R¬ MäEä*ã\ä � �C®0è>ç$¯Mã\ä�é+çæê 3ê ª � åPç�ê��ã�âDè>ç�â �<ã��3ê 7å���è � ���¬ Mä*âEã_å � è � � ���ç5è f

¡M4�Aa1¢a2¢. . .¢anJ34aID476M2r:B?<L�J34MI�8\¢<476 i GN¥�ID4767AD4MI�A+GB6M:

1f 1<ID=M�B4OA+GB67A

n∑

k=1

(

1 + logak(ak+1)

)2 ≥ 4n¢

� G34MID4an+1 = a1

fáO¯ ë � ftª ä � � � é�ã � ���`²�ç�ê ���"¯ � è Mä ��� � � ± «Mèyã��é�êKç � � � ±�MèBç �� Mè �5¬ � � � Mè � è>çæê ��� �¬ �Bâ ���ä+ã\é�ê��� � ±�MèBç f

� ka¢b¢B6�:<;

c6MID4aH<=B8*C�ADC��B4�ID476M2r:B?<L�J34MI�8\¢3H3ID=M�B4RA+GB6�A

a2(b + c − a)

b + c+

b2(c + a − b)

c + a+

c2(a + b − c)

a + b≤ ab + bc + ca

2

fáO¯ ë © ftª ä � � � é�ã � ���`²�ç�ê ���"¯ � è Mä ��� � � ± «Mèyã��é�êKç � � � ±�MèBç �� Mè �5¬ � � � Mè � è>çæê ��� �¬ �Bâ ���ä+ã\é�ê��� � ±�MèBç f

¡M4�Aa¢b¢B6�:<;

cJ<4OA+G34�2K4M:<¥_A+GB8$=>krA+G34a8+C5;34M8$=>k 6�AEI�CK6M:<¥\254 f 1<ID=M�B4OA+GB67A

(b + c)2

a2 + bc+

(c + a)2

b2 + ca+

(a + b)2

c2 + ab≥ 6

fáO¯ í ± f ª ä � � � é�ã � � � �~µ ¶Jµ � ±�ãMã�è � ���åæê � � ±�±�ã-å ��ê� ã`´aã�ê� ã�ä�å �è���é f

¡M4�AABC

J34R6M:�CK89=B8 i 472K4M8�AEI�CK6M:<¥\254$��C�A+GAB = AC

6M:<;∠BAC = 100◦ f¡M4�A

DJ34FA+G34�H<=>CK:>Aa=B:SA+G34QH3ID=3;B? i ADCK=B:w=>k

AB8+? i GSA+GB67A

AD = BCf � CP:<;

∠ADCf

� �3�

�h� � � ��� � �

´ � �yä � �Må�ã�±¿çKé�ã3¯Mã�ä���ã�äU±�Mèyã\è�êKå �Fâæå � é\ã���µ�² �yã"ã ��çæê � ä�çPé��å$­ ��Mé �rå�ã Mé\ã��ê � â � èBé*ç �<ã�ä � � ä�� � �MåPç�â�BêKç � èNèyãJ­né � å �BêKç � èBé � äzèyãJ­jçKè3é+ç � ��êPé � è���7é7ê �yä � �Må�ã�±NéJµ

á Ø á � f �D�M�����Vì � �M� ¢ � � � � ª ä � � � é\ã�� ��� ª �7åE¬ ä�Bâ�Bã�è �`®0è>ç$¯Mã\ä�é+çæê � ��� ²Mã� Mé �� ��çKè � �Mä � ��²�����® � � f

¡M4�Aa1J34�6F:<=B:<�EÞ*47ID=VID476M2�:B?<LFJ<47I-¢�6M:<;�;<4��r:<4�A+G34F8-4M�B?r47: i 4 {an}∞

n=1JMTan+1 = n2/an

ku=BIn ≥ 1

f 1<ID=7�B4RA+GB6�AN∑

n=1

1

an

=

(

1

πa1

+πa1

4

)

ln(N) + O(1)f

��� å��3êKç � è ���t¶ � ã-å � â �>å � é��3ã�ä ����¬ ��Mé+ç��<ã ��´ ����® � ����± � ��ç��yã�� � ��ê�yã ã ��çæê � ä f� A�CK8�8EAEI�6MCK¥7G�ADku=BIæ�a6MID;"AE="H<ID=7�B4 JMT�CP:<;B? i ADCK=B:�A+GB6�A�ku=BI

n ≥ 0¢

a2n+1 =

(

22nn!2

(2n)!

)2

a1f

g�i+i =BID;BCP:<¥XAE= � ADCPID2@CK:<¥�§ 8 � =BIEL�?32K6B¢n! =

√2πn

(

n

e

)n[

1 + O(

1

n

)] ku=BIn → ∞ f

� G347ID47ku=BID4>¢Bku=BIn ≥ 1

¢

22nn!2

(2n)!=

22n2πn(

n

e

)2n

√

2π(2n)(

2n

e

)2n

[

1 + O

(

1

n

)]

=√

πn

[

1 + O

(

1

n

)] f

� G347:y¢Bku=BIn ≥ 1

¢1

a2n+1

=1

πna1

[

1 + O

(

1

n

)]

=1

πa1

[

1

n+ O

(

1

n2

)]

6�:<;1

a2n+2

=a2n+1

(2n + 1)2=

πna1

4n2(

1 +1

2n

)2

[

1 + O

(

1

n

)]

=πa1

4n

[

1 + O

(

1

n

)]

=πa1

4

[

1

n+ O

(

1

n2

)] f

� � �

� G347ID47ku=BID4>¢Bku=BIN ≥ 4

¢N∑

n=1

1

an

=1

a1

+1

a2

+

bN−1

2 c∑

n=1

1

a2n+1

+

bN−2

2 c∑

n=1

1

a2n+2

=1

πa1

bN−1

2 c∑

n=1

[

1

n+ O

(

1

n2

)]

+πa1

4

bN−2

2 c∑

n=1

[

1

n+ O

(

1

n2

)]

+ O(1)

=

(

1

πa1

+πa1

4

)

lnN

2+ O(1)

f� =�ADCP:<¥�A+GB67A

ln(N/2) = ln N − ln 2 = ln N + O(1)¢�� 4R6MIEI�C��B4R6�A�A+G34a;34M8+CKID4M;ID4M8+?32@A f

��� ���l���������a ��dNd�R���H 3D o �EI2���+D#DB �$J�9�3Q#�(B$G=�1R&�(')�h8�����DBIB�! [������CH<i57"�$g5'1��.Q�� S (��(��p���FZ(3&���S+Q�FJ$�� (3(#�� �1LQ#)+q�$3��Q���P 1RS &�8'&�('�ZP *'E]�12�.]K�����1��

�������� ��������������������������! #"%$�&'$�(�)�*,+�-/.1032546)87:9<;�=>;?0@7A7B)DCFE'$#G�)�H�C:I�"�;�H<2J)��K�L?M

ε N LOMJP�LQL?R@RASUT�V8L x2

a2+

y2

b2−1 = 0

�XWZY�[ T�\�]�\�RAR>L?R�M�\�^�_�L�^ZM5VXM [εS@^ZM`L�]�V8L�a5M

\bMJPZS@]5c!M5\�^�_�L�^ZMO\�MM1(x1, y1)

\�^�cM2(x2, y2)

�Fd P [�Y MJP�\?MFMJP�L�eZ\�R@f#L [Zg(

x21

a2+

y21

b2− 1

)(

x22

a2+

y22

b2− 1

)

SUVhS@^�c�L�T#L�^�c�L�^ZM [Zg MJP�L�a%P [ ViL�^!M�\�^�_�L�^ZM5V �j 2`$�kl+�0>H�;�=U03$�Hm$�no(J0>k/0A7>;�"p(�$�7�G�=U0B$�H�(�+�- j &'$�(�=3$�7@0@(�qorOs�)�k/0U(%Cut�;�"%v�;�w�)80B$yxp0Uz�4{ 254<$�$�7UC j =|4<)?H�(iC:}X"%)�)�2`)D~6;�H<*, #)�=B)�"���rF�!$�$�C:9�03$�7>;l�XH�03v�)?"�(%0B=|-�CF��;!.y0>"�;�*Z;�CF� j C� { j �

� L�V8P�\�R@R'V8L�L!MJP�\�M g|[ ]p\�RARFa%P [ SUaJL�V [Zg MJP�L/MJP�]5L�L!M5\�^�_�L�^ZM�V � MJP�L�T�] [ c�f#a5MpS@^� f#L�VJM5S [ ^!SUVh\�R Y \���V

1� � Lbc�L�^ [ M5L N � O

MJP�L�a`L�^ZM`]5L [Zg MJP�Lb_�S�e�L�^�L�RAR@S@T�ViL � \�^�c N �C1

�C2

� \�^�cM

MJP�L�T [ SU^ZM�V [Zg M�\�^�_�L�^�a�� [Zg MJP�L�MJP�]5L�L�M5\�^�_�L�^ZM�VC1M1

�C2M2

�\�^�c

M1MM2

�W P�L�\��,^�L�M`]�\�^�V g|[ ]`��\�M5S [ ^

(x, y) →(

x,a

by) M`]�\�^�V g|[ ]5��V�MJP�L�_�S e�L�^

L�RAR@S@T�ViLM [ MJP�Lua`SU]5a`RUL Y S�MJP!a`L�^ZM`]5LO\�^�c!]�\�cZS@f�V

a�o� MF\�R@V [ M`]�\�^�V g|[ ]`��VoMJP�L�T [ S@^ZM5V

M1(x1, y1)�

M2(x2, y2)�

C1

�C2

� \�^�cM

M [ MJP�L�T [ SU^ZM�VM ′

1

(

x1,a

by1

) �M ′

2

(

x2,a

by2

) �C′

1

�C′

2

� \�^�cM ′

� ]5L�V%T#L�a5M5S�e�L?R�� � \�^�c�S�M'M`]�\�^�V g|[ ]`��VoMJP�LM5\�^�_�L�^ZM�VC1M1

�C2M2

� \�^�cM1M2

[Zg MJP�L�L?R@RASUT�V8L�M [ MJP�LuM5\�^�_�L�^ZM�VC′

1M ′1

�C′

2M ′2

� \�^�cM ′

1M ′2

� ]5L�ViT�L�a5M5S e�L�R � ��[Zg MJP�L�a`S@]5a`R>L �� [ ^�ViL � f#L�^ZM5R � �M ′

1O ⊥ M ′2O �>N S@V8L�a5M [ ]�V [Zg ∠C′

1OM ′ \�^�c∠M ′OC′

2

�Y P�L�]5L∠C′

1OM ′ + ∠M ′OC′2 = π � � \�^�c OM ′ ⊥ M ′

1M ′2 � V%S@^�aJL M ′

1M ′2

S@V

� �3�

AD6M:<¥74M:>A<AE=$A+G34 i CKI i 2K4z6�:<;OM ′ CP8�6�I�6�;>CP?<8Dà f � 4 i 67?<8-4 4M ′

1M ′O ∼ 4OM ′M ′2

¢C@A�ku=>2@25=��a8�A+GB67AM ′M ′2

1 · M ′M ′22 = M ′O4 f�� G34M:

(

M ′1O2 − M ′O2

)

·(

M ′2O2 − M ′O2

)

= a4 ¢((

x21 +

a2

b2y21

)

− a2

)

·((

x22 +

a2

b2y22

)

− a2

)

= a4 ¢(

x21

a2+

y21

b2− 1

)

·(

x22

a2+

y22

b2− 1

)

= 1f

��� ��� ���������f ������!�R,a��59��IB�U"�$MNeS �2���3Q�1RSE"#�3QBPU*��1�(l"BP &BP+�5'(#SU���1R��SUPU�.$��.�.][� S (#$iN%<9$J5K"L��8�M�ED#I��H 3�����ECH<@5K".$ 5'1��.Q�� S (��(#p���FZ(#&���SQ�FJ$G� (3(#�� �1LQ#)+q�$H�MQ���PU1RS &.8T�������������-���!�! #"�"�$J�'&�(�)R*��,#&��U���\/'�12�3���$@����$X57"��98���<> [DG"#���HD#<�" o [��:�$ o &�����S�3L$�CH?3$X57"��98��i� ���9"BNd 3 3C!A�$>k.&.�P+ 3��FZFGL� $PU*�iC!�PU*��1R� &�('����8�&�(��ZPU*�E]�12�.]K�����1��

���������� �������������������������� #"%$�&'$�(�)�*,+�-/.y0B254<)D7:9<;�=>;?0@7A7B)DCFE'$#G�)�H�C:I�"�;�H<2J)��K�L?M

a1

�a2

�. . .

�an N L n

cZSUV`M5SU^�a5M�T [ VJSAM5S e�L�]5L�\�RX^�f�� N L�]�V ��Y P�L�]5L n ≥ 2�

� [ ]i = 1

�2�. . .

�n� RUL?M

pi =∏

j 6=i

(aj − ai)�od P [�Y MJP�\?M n

∏

i=1

a1

pi

i < 1�

{ $�7�G�=U03$�H +�- ��;?7�=|4<)?"��8;�H<$#G�(%C ��"5(�G�7A0>H<)?H�z�-�k�H�;�(%0�G�k�C��3H�H�(?+�"DG�25w�C j G�(?=|"�0>;�Ck $�*�06)�*,+�-/=|46)�)�*�0B=B$�"D�� L Y SAR@R�\�T�T�R �pMJP�L g|[ RAR [�Y S@^�_FMJP�L [ ]5L�� [ ^ba [ ^�e�L�� g f�^�a5M5S [ ^�V [Zg PZS>_�P�L�] [ ]5c�L�]

� ViL�L � Z� T�T �Z������� � �������������� � K�L�MI N L�\�^ [ T�L�^�SU^ZM5L�]3eZ\�R � \�^�c�RUL?M f : I → � N L�\ g f�^�a5M5S [ ^ Y PZSUa%PSUV

n � M5SU��L�VQc�S���L�]5L�^ZM5S@\ N R>L �'W P�L g|[ R@R [�Y SU^�_�VJM�\�M5L���L�^ZM�VQ\�]5L�L � f�S eZ\�RUL�^ZM �

� S � �|g x0

�x1

�. . .

�xn

\�]5LQ\�^��n+1

cZSUV`M5SU^�a5MXT [ S@^ZM5V'S@^I� MJP�L�^ n

∑

i=0

f(xi)

w′(xi)> 0

�Y P�L�]5L

w(x) =n∏

j=0

(x − xj)�

� SAS � f (n)(x) > 0g|[ ]o\�RAR

x ∈ I�

g f�^�a5M5S [ ^fVi\?M5SUV g ��S@^�_pMJP�Lb\ N [ e�L�a [ ^�cZSAM5S [ ^�V�S@V�Vi\�S>c�M [ N L (�=|"�032i=U7|- n ! 2`$�H6v�)#"<�� [ ] [ f�]pT�] [ N RUL�� �6Y L�M5\�$�L I = (0, ∞)

\�^�cf(x) = (−1)n ln x

�/W P�L�^f (n−1)(x) = (n − 2)!/xn−1 > 0

g|[ ]�\�R@Rx > 0

� W P�L�]5L g|[ ]5L �fS@V!VJM`]�S>a5M5R �

(n − 1) � a [ ^�e�L%� �&QV%S@^�_�MJP�LncZSUV`M5SU^�a5MO^�f�� N L�]�V a1

�a2

�. . .

�an

_�S�e�L�^�SU^!MJP�L�T�] [ N R>L�� ��Y LR>L�Mw(x) =

n∏

j=1

(x − aj)�'W P�L�^ g|[ ]hL�\�a%P

i = 1�2�. . .

�n��Y L�P�\�e�L

w′(ai) =∏

j 6=i

(ai − aj) = (−1)n−1pi

�

� ���

g�i+i =BID;BCP:<¥OAD= i =B:<;>C@ADCK=B:FßPC�à�CP:"A+G34RA+G34M=BID4ML�¢ n∑

i=1

f(ai)

w′(ai)> 0

f�� G34M:n∑

i=1

ln ai

pi

= −n∑

i=1

(−1)n ln ai

(−1)n−1pi

= −n∑

i=1

f(ai)

w′(ai)< 0

f

� 6 ¤ CP:<¥a4+ò�H<=B:<4M:>ADCP6M2P8�¢�� 4a=3J�AD67CK: n∏

i=1

a1

pi

i < 1f

� ï_óæï �*ï�î �_ï7ö� � � ��C@AEI�CK:<=7�BC i ¢ ÙRf �<f ¢ �>å Mé*é*ç�â��å Mè � ´aãJ­ uèyã� � �åPç�êKç�ã_éQçKè ��è�7å �Mé+çKé ¢ �y2P?>� 47Ig�i 6�; f 13?rJ>2 f ¢ Ù =BID;BID4 i G�A-¢ �����>� f

��� ���!���������Z ��#��<> [D!"3���!D3<�" o 3��:�$ o &�����SU�3L$7CH?3$E5K"L��8�&�('�ZPU*�E]�12�.]K�����1��

�����b���/ 5������� ���������6�?����� #"J$�&'$�(?)�*l+�- �?$�( �)/��G�0@(�s �� ;�� ! 9<;�"`"%)�"%$�C���HZ0Bv�)�"�(J0�=U;�= #$�7@0B=��)�25HZ0B2�;�*�)���;�=U;�7�G�H�-�;�CF9<;�"%2`)D73$�H�;�C { &6;?0>HX�K�L?M

a�b� \�^�c

c N L�MJP�L�VJS>c�L�V [Zg \�Via`\�R>L�^�LuM`]�SU\�^�_�R>L ABC��� ] [ e�LuMJP�\?M

∑

� � �

(a + 1)bc(√

a −√

b)(√

a − √c) <

a4 + b4 + c4

abc

�

{ $�7�G�=U03$�Hl+�-/.y0B254<)D7:9<;�=>;?0@7A7B)DCFE'$#G�)�H�C:I�"�;�H<2J)DCXk $�*�06)�*,+�-�=|4<)�)�*�0B=B$�"D�� ]5S�M`L�MJP�LbRUL g MoV%SUc�Lb\�V

S1 + S2

��Y P�L�]5LS1 =

∑

� � �

abc(√

a −√

b)

(√a − √

c)

�S2 =

∑

� � �

bc(√

a −√

b)

(√a − √

c)

�

� N ViL�]3e�LuMJP�\?M

S1 = abc ·

(√c −

√b)

+(√

a − √c)

+(√

b − √a)

(√c −

√b)

(√a − √

c)

(√b − √

a) = 0

�

� L�a5\�^�VJSU��T�RAS g �S2 N ����\�$�SU^�_�f�V8L [Zg MJP�LuS>c�L�^ZM5S�MB�

x2y2(y−x)+y2z2(z−y)+z2x2(x−z) = (x−y)(y−z)(z−x)(xy+yz+zx).

�OS�eZS>cZSU^�_ N [ MJP/VJS>c�L�V N � (x − y)(y − z)(z − x)\�^�c/ViL?M�M5S@^�_

x =√

a�y =

√b�

\�^�cz =

√c��Y L [ N M5\�SU^

S2 =√

ab +√

bc +√

ca�

W PZf�V � MJP�L�_�S�e�L�^�S@^�L � f�\�RASAM3�!M5f�]5^�V [ f�MFM [ N L�L � f�S�eZ\�R>L�^ZMFM [abc

(√ab +

√bc +

√ca)

< a4 + b4 + c4� � �

� � �

� =���¢>k5ID=BL A+G34��S4�C5¥7G�AD4�; g � 4�� � :<4M�B?<672PC�A�T ¢�� 4aGB6\�B4a

3

2 b3

2 c = (a4)3

8 (b4)3

8 (c4)1

4 < 38a4 + 3

8b4 + 1

4c4

ß@� G347ID4OA+G34OCP:<4��>?<6M2@C@A�T�CK8z8EAEIDC i A�J<4 i 67?<8-4a¢b¢>6M:<;

c6�ID4�;>CK8EADCK: i Aæà f�� CPLNC@2K6MID2 Ty¢7� 4GB6��B4

a3

2 bc3

2 < 38a4 + 1

4b4 + 3

8c4 6�:<; ab

3

2 c3

2 < 14a4 + 3

8b4 + 3

8c4 f � k � 4a6�;3;6M2@23A+GBID4�4�CP:<4��>?<6M2@C@ADCK4M8\¢�� 4a=3J>A�6MCP:Fß � à f� =�AD4FA+GB6�AOA+G34�ID4M8+?32@A�6 i AD?<6M2@2�T�G3=>2K;38R� G347:<4-�B4MI

a¢

b¢�6M:<;

c6MID4Q;BCP8+ADCP: i AHr=B8+C@ADC �B4�ID4M672 :B?<LFJ<47I�8 f

��� ���M���������H ��>����A�����?i�ED#I2$["�&�(<������L$7����$�57"��987���!�R,j��5���I��".$�NeS �2���3Q�1RS#"#�3QBPU*��1�(X"BP &BP+57(BS���1R��SP ��$��.�.][� S (#$�N%<9$H57"��98��������������0���>�! #"�"�$��'&�(')+*'�T,#&.����Z/'�12�3���$@����$H5K"L��8�����DBIB�! [�����CH<i57"�$Z5'1��.Q�� S (��(#p���FZ(#&���SQ�FJ$!�U(#(3�� �1LQB)Rq�$���Q���P 1RS &�86��<� 3DT"#���HD#<�" o [��:�$ o &�����S�3L$!CH?3$J5K"L��8,7 7I� [�f?9���J<E<K$ o S��� &g5'(#SU���1R��SUPU�.$KD�&XNeS 1R&B��&�$9����$E57"��98'&�('�ZP *'E]�12�.]K�����1��

"#����1R&�����������1R��FG�(�P SU��('��!PU*#&#P'PU*��12��.Q.�PBS ��P 1LQ#��+��1K&�([�!P *312����S ��P S (')�PB]K����SP S���12�&.�B(7Q�F� 3�1R���JNd����P���������1R�! 3Lp�&�(#$9� S q3 o &BP &.S � �L$� ��J�2*��B^�S (BpGP *3&BPEP *'ip�SU���(jS ('��[Q�&�� SP �XS �H��[Q.S�L&��U�('P�P�abc(

√ab +

√bc +

√ca) < a4 + b4 + c4

�'�EPKPU*#&#P3]K��S ('P $KP *'�12M^@�12�������1R&.��^E&����EP+�H)���FX][�U�P+PU*�M�����+QBP SU��(K�?�h&�(��3Q���]�12�B�LS�3��!������1R&.�[�.QB)R*j^E&�����*BS FZ��L� ����������u�� ���������������������� ?� #"%$�&'$�(�)�*,+�-/.y0>4 �;?7|-�9�)�H62 �D)DCF9<"�;�(�$�v�CFE'$�k�;�HZ0>;��

�|gxk > 0

g|[ ]k = 1

�2�. . .

�n� T�] [ e�L�MJP�\�M

� \ � cos

nn∑

k=1

xk

− sin

nn∑

k=1

xk

≥ 1

n

n∑

k=1

(

cos1

xk

− sin1

xk

) �

�>N��n∑

k=1

sin1

xk

n∑

k=1

cos1

xk

≥ tan

nn∑

k=1

xk

�

� *�0B=B$�"� (�2J$�k�k )�HX=5�&Q^ g|[ ]�M5f�^�\?M`L?R�� � MJP�L�SU^�L � f�\�R@S�M5S>L�V�\�]5L�SU^�a [ ]`]5L�a5Mu\�VbVJM�\�M5L�c ��<[ MJP � \�R�MJP�L�]

�%\�^ [ f�V � &Q]5V%f�RASU^�L�^�_8����^�\�VJS@f�� �� ^�^�V N ]�f#a�$ � f�VJM`]�SU\ \�^�c � L�M`L�]� � � [�[<�� S [ RU\&Q^�S�e�L�]�V%S�MB� � K�\��yS@]�\�c�\ � � � & d _?\�e�L�a [ f�^ZM`L�]5L���\���T�R>L�Vb\�^�c MJP�L�^l\?M�M`L���TZM`L�cM [ SU��T [ ViL�\�c�c�S�M5S [ ^�\�RF]5L�VJM`]�S>a5M5S [ ^�V [ ^ MJP�L/eZ\�]5S@\ N R>L�VuM [ ��\�$�L/MJP�L�SU^�L � f�\�R@S�M5S>L�Va [ ]5]5L�a5M � �%\�^ [ f�V�V%f#a`aJL�L�c�L�clSU^�]5L�T�\�SU]�SU^�_�T�\�]�M � \ � � � [�[�� V�a [ f�^ZM5L�]5L%��\���T�RUL g|[ ]T�\�]�M � \ � SUV n = 2

�x1 = 0.4

� \�^�cx2 = 0.3

� \�^�c��%\�^ [ f�V � a [ f�^ZM`L�]5L���\���T�R>L g|[ ]T�\�]�M �|N�� SUV n = 2�x1 = 1/π

� \�^�cx2 = 3/π

�{ $�7�G�=U03$�H =3$ ;Z*��8G�(�=3)�* &6;�"8= � \ � +�- ��;?7�=|4<)?" �8;�H<$#G�(%C��X"5(�G?7@0UH6)�H�z�-�k�H�;�(J0�G�k�C�3H�H�(?+�"DG�25w�C j G�(?=|"�0>;��

K�L?Mf(x) = cos

1

x− sin

1

x

�O� ^�L � f�\�RASAM3� � \ � a5\�^�^ [�Y N L Y ]�SAM�M5L�^�\�V1

n

n∑

k=1

f(xk) ≤ f

(

1

n

n∑

k=1

xk

) �W PZS@V�S@V��iL�^�ViL�^ � V � ^�L � f�\�RASAM3� a%P�\�]�\�a5M5L�]�S��`S@^�_lMJP�Lya [ ^�a5\?eZSAM3� [Zg MJP�L g f�^�a5M5S [ ^

f�

� � �

Ú 8*CP:<¥R6 i =BLFHB?BAE47Iz6M2K¥74�JBI�6 8�TB8+AD4ML�¢�� 4 i 6M:F89G3=���A+GB6�A

f ′′(x) =(1 − 2x) sin

1

x− (2x + 1) cos

1

xx4

< 0

ku=BIx ∈ (0.601451551, ∞)

f ß � G34a4M;BC�AE=BI0�B4MI�CP�r4�; �*6�:<=>?<8\§�ID478*?32�A�?<8*CP:<¥ �w6�HB254 f à� G>?<8\¢BCP:<4��>?<6M2@C@A�TFßK6�à�CP8zAEID?r4RCPkxk ∈ (0.601451551, ∞)

ku=BIk = 1

¢2¢. . .¢nf

��� å��3êKç � èSê � ��� �Mé�ê�ã � ��Mä-ê ß5J�à � ��ê�yãaã���ç�ê � ä �Mé+ç5è �Rç��<ã Mé ��� ê� ã �yä � � � é�ã\ä f¡M4�A

g(x) = sin1

x

6M:<;h(x) = cos

1

x

f�� G347:g′′(x) =

2

x3cos

1

x− 1

x4sin

1

x

6�:<;h′′(x) = −

(

2

x3sin

1

x+

1

x4cos

1

x

) f� =BI

x ∈ (0.928613759, ∞)¢�� 4QGB6\�B4

g′′(x) > 06M:<;

h′′(x) < 0f � :�k56 i A-¢

h′′(x) < 0CPk

x ∈ (0.436885409, ∞)f�� CK: i 4

0.928613759 > 2/π¢�� 4 GB6\�B4

0 <

nn∑

k=1

xk<

π

2

6�:<;0 <

1

xk

<π

2

f� =���¢ �947:38-47:y§ 8 � :<4��>?<6M2@C@A�TN6�H3H32@C54M;"AE= A+G34�kK?<: i ADCK=B:38

g(x)6�:<;

h(x)T>C54�25;B8

1

n

n∑

k=1

sin1

xk≥ sin

nn∑

k=1

xk

> 0

6M:<;0 <

1

n

n∑

k=1

cos1

xk≤ cos

nn∑

k=1

xk

¢k5ID=BL � G>C i G�A+G34RCK:<4M�B?<672PC�A�T"CK:NH36�IæA�ßuJ�à�ku=>2@25=��a8zCPLFLF4�;>CK6�AE4�2�T f�����!�u�� ���������������������� ?� #"%$�&'$�(�)�*,+�-/.y0>4 �;?7|-�9�)�H62 �D)DCF9<"�;�(�$�v�CFE'$�k�;�HZ0>;��

� [ ]n ≥ 1

R>L�Man = 1 +

√2 + 3

√3 + · · · + n

√n��� ] [ e�LuMJP�\?M

n∑

k=1

k√

k

a2k

<2n + 1 + (ln n)2

n + 1 + 12(ln n)2

�

{ $�7�G�=U03$�H,+�-�E60B254�;�"J* �5r#x�)�(%(iC�E6;�H<2546$! �;�73$�(ut#)?"J*�)�(%C'� j Ch� { j C�k $�*�0 X)�*/+�-�=|4<))�*�0B=B$�"D���L���^�L

Ln =

n∑

k=1

k√

k

a2k

\�^�cRn =

2n + 1 + (ln n)2

n + 1 + 12(ln n)2

� �uc�S@]5L�a5MXa [ ��T�f�M5\?M5S [ ^ �8Y LhViL�LFMJP�\�M

L1 = 1�L2 ≈ 1.2426

�L3 ≈ 1.3396

�L4 ≈ 1.3905

�R1 = 1.5

�R2 ≈ 1.6914

�R3 ≈ 1.7828

� \�^�cR4 ≈ 1.8322

���L�^�aJL �

Ln < Rn

g|[ ]n = 1

�2�3�4�

� � �

� =�� 6M898+?<L�4OA+GB67An ≥ 5

f � CP: i 4OA+G34�kK?<: i ADC5=B:f(x) = x1/x CK8$;34 i ID476�8+CK:<¥=B:

(e, ∞)¢B� 4�GB6��B4>¢<ku=BI$4M6 i G

k ≥ 5¢1 < k1/k ≤ 51/5 ≈ 1.3797

f � =�AE4 6M2P8-=A+GB67Aak ≥ k

ku=BI�6M2@2k ≥ 1

f�� 47: i 4>¢

Ln = L4 +

n∑

k=5

k√

k

a2k

< L4 + (1.3797)

n∑

k=5

1

k2< L4 + (1.4)

∫ n

4

dx

x2

= L4 + (1.4)(

1

4− 1

n

)

< (1.4) + (1.4)(

1

4

)

= 1.75f

� CK: i 4Rn = 2 − 1

n + 1 + 1

2(ln n)2

> 2 − 1

n + 1≥ 2 − 1

6> 1.8

¢�C@A�ku=>2@25=��a8A+GB67A

Ln < Rnf

��� ���!���������Z ��h�M�ED#I��H 3�h���ECH<@5K".$E5'1��.Q�� S (��(#p���FZ(#&���SQ�FJ$K�U(#(3�� �1LQB)Rq�$3�MQ���PU1RS &[��L&�('�3Q����������Q#P S��(B$G^E*BS)+*;S �e �&�����a��(6)���FX]9Q#P�1 ���1RS V7)&#P S��(#��$\&B)2PRQ�&.� � �;���P &B �� S �2*'��TPU*���PU12��(Bp#�1XS (���[Q�&.� SUPU� n

∑

k=1

k√k

k2< Rn

� o �#P *u�L&�('�3Q��j&�(��%�!��2� 3L� S���TPU*#&#PXP *'gFJS (BS FgQ�F � �Rn − Ln

S �i&BPRP &�S ('��g^E*��(n = 8

8MPU*'Q���$Ln ≤ Rn − 0.4409

^E*'�120.4409 ≈ R8 − L8

$ [Q#P9P *'��@p�&#��M(��@]�12�#� ���EI�*�M�����Q#P S��(j ��XP *'�]�12�.]K�����1�&.]3][� S��!��S FX][�iPL����)��.][S (BpXP��p#�P *'�1>^�SUPU*PU*�@S (���[Q�&.� SUPU�

an < n + 1 + 1

2(ln n)2

$�^E*#SU)R*\*'H)+� &�S FG��j)&�( 3@�2*'�B^E(g ��!S (��[Q#)�P SU��( [Q#P���SU�('�#P7�.Q�]3][� �i&�([�@]�12�#� ���

����� �/�� ���������������������� ?� #"%$�&'$�(�)�*,+�-/.y0>4 �;?7|-�9�)�H62 �D)DCF9<"�;�(�$�v�CFE'$�k�;�HZ0>;��K�L?M

A�B ∈ Mn(

�)� \�^�c/RUL?M

ω = e2πi/n� � ] [ e�LuMJP�\?M

n∑

k=1

det(A + ωk−1B) +

n∑

k=1

det(B + ωk−1A) = 2n(det A + det B)�

� *<r�W P�L�T�] [ N R>L�� P�\�V N L�L�^ a [ ]5]5L�a5M5L�c/M [ VJM�\�M5LbMJP�\�M ω = e2πi/n� \�V�MJP�LT�] [ T [ V8L�]FS@^ZM`L�^�c�L�c � ]�\�MJP�L�]:MJP�\�^

ω = e2π/n��W PZS@V�a [ ]5]5L�a5M5S [ ^ Y \�Vh��\�c�L N ��MJP�LV [ R e�L�]�V � �

{ $�7�G�=U03$�Hl+�-/.y0B254<)D7:9<;�=>;?0@7A7B)DCFE'$#G�)�H�C:I�"�;�H<2J)��� [ ]

M ∈ Mn(�)�#Y L�c�L�^ [ M5L N � M (1)

�M (2)

�. . .

�M (n) MJP�L�a [ RAf���^�V[Zg

M� K�L�M

x N Lu\�^�SU^�c�L?M5L�]5��S@^�\�M5L �Fd S@^�aJLu\�^ n × nc�L?M5L�]5��S@^�\�^ZM:SUVO\�^

n� RASU^�L�\�]g f�^�a5M5S [ ^ [Zg SAM�VQa [ RAf���^�V ��Y L�a`\�^,L%��T�\�^�c

det(A + xB)\�V g|[ RAR [�Y V �

det(A + xB) = det(

A(1) + xB(1), A(2) + xB(2), . . . , A(n) + xB(n))

= det(

A(1), A(2), . . . , A(n))

+

n−1∑

j=1

αjxj

+ xn det(

B(1), B(2), . . . , B(n))

= det A +

n−1∑

j=1

αjxj + xn det B

�

� � �

� G34MID4α1¢α2¢. . .¢αn−1

6�ID4 i =BLNH32K4*òF:B?<LFJ<47I�8aßPCK:<;34MH<4M:<;34M:>AR=>kxà f�� 6 ¤ CK:<¥

x = 1¢ω¢ω2 ¢ . . .

¢ωn−1 CK:N8+? i+i 4M8*8*CK=B:N6M:<;N6�;<;>CK:<¥XA+G34aID478*?32�AD8\¢�� 4 =3J�AD67CK:

n−1∑

k=0

det(A + ωkB) = n det A +

n−1∑

k=0

n−1∑

j=1

αjωkj +

n−1∑

k=0

ωkn det B

= n det A +

n−1∑

j=1

αj

n−1∑

k=0

ωkj +

n−1∑

k=0

ωkn det Bf

� =���¢�8+CK: i 4ωn = 1

6M:<; n−1∑

k=0

ωkj =1 − ωnj

1 − ωj= 0

ku=BIj = 1

¢2¢. . .¢n − 1

¢� 4 GB6\�B4

n−1∑

k=0

det(A + ωkB) = n(det A + det B)f

� G347: n−1∑

k=0

det(B + ωkA) = n(det B + det A)¢B6M:<;"A+G34aID478*?32�A�ku=>2P2K=��a8 f

��� ���@�����U����H ��!,7 7I� [�T?9�?�J<E<K$ o SU��� &\57(BS���1R��SP ��$�D�&iNeS 1R&B��&�$K����$>57"��983&�(��XPU*��]�12�.]K�����1��

������� ? � ����������O��� ���p�?� �� #"%$�&'$�(�)�* +�- �#G�k/0UH�z���4<)?H�C,�u0A7Un>"�0B* ��;�G�"�0B)�"��HZ0Bv�)�"�(J0�=>-�Ch��;�=3)?"�73$�$�C������K�L?M

f(x) = x − c2 tanh x�XY P�L�]5L

c > 1S@V�\�^y\�] N SAM`]�\�]3�ya [ ^�V`M5\�^ZM �1� MSUVQ^ [ M�P�\�]5c/M [ V8P [�Y MJP�\�M

f(x)S@Vc�L�aJ]5L�\�V%S@^�_ [ ^/MJP�L�S@^ZM`L�]3eZ\�R

[−x0, x0]��Y P�L�]5L

x0 = ln(

c +√

c2 − 1) S@VQMJP�L�T [ VJSAM5S e�L�] [�[ M [Zg MJP�L/L � f�\?M5S [ ^

cosh x = c� � [ ]L�\�a%P

x ∈ (−x0, x0)� MJP�L�P [ ]5S � [ ^ZM5\�R#R@S@^�L�T�\�ViVJSU^�_�MJP�] [ f#_�P (

x, f(x)) S@^ZM`L�]�V8L�a5M5VMJP�L�_?]�\�T#P [Zg

f\�M:M Y�[/[ MJP�L�]oT [ SU^ZM�V Y SAMJP!\ N Via`SUV%Vi\�V x1(x)

\�^�cx2(x)

� ��L �#^�Lb\g f�^�a5M5S [ ^g : (−x0, x0) → � \�V g|[ RAR [�Y V �

g(x) = x + c2 tanh(

x1(x))

+ c2 tanh(

x2(x))

�� ] [ e�L [ ]hcZSUV%T�] [ e�LuMJP�\?M

g(x) > 0g|[ ]o\�RAR

x ∈ (0, x0)�

� *�0B=B$�"� (oH6$�=3)����[ V [ RAf�M5S [ ^�V Y L�]5Lp]5L�aJL?S�e�L�c g|[ ]6MJPZSUVFT�] [ N RUL��� P�L�^�aJL � SAM']5L���\�SU^�V[ T#L�^ �'W P�LuT�] [ T [ ViL�] N L�RAS>L8e�L�VoMJP�\�M'MJP�Lba [ ^�3L�a5M5f�]5LuS@VoM`]�f#L � VJSU^�a`LpMJP�L�]5LuS@Vh\���T�R>LL���T�S@]5SUa`\�R6L8eZS>c�L�^�a`L �

� �������������������������! ���"#�%$'&)(+*-,*�.%/�0�1�2�3!4-.65879/;:)<= (?>@*�4BA+/�CEDGFH59I�/�(J.+5K>L2M*�N#&O79*�5@/�.%>P5PCQ *�RS4�F!5P4#�T�U%V

p > 1 !W�X#Y[Z U%V

a b c d ∈

[

1/√

p,√

p]

� \#]�^-_ UMV+` W Va W�b p

1 + p+

2

1 +√

p≤ a

a + b+

b

b + c+

c

c + a≤ 1

1 + p+

2√

p

1 +√

p caLd b p

1 + p+

3

1 + 3√

p≤ a

a + b+

b

b + c+

c

c + d+

d

d + a≤ 1

1 + p+

3 3√

p

1 + 3√

p

�

� �B�

��� å��3êKç � è ������ä ������å�ê� � Mè`¶ � é\ã ��������® � ����± � ��ç ��ã � � ��ê�yã ã ��çæê � ä fß56\à � G34OAEI�6�:38*Hr=B8+C@ADCK=B:

(a, b, c) 7→ (b, a, c)CP:"A+G34�CP:<4��>?<6M2@C@A�T

a

a + b+

b

b + c+

c

c + a≤ 1

1 + p+

2√

p

1 +√

p

ß � à¥\C��B478�A+G34a4��>?3C��>672547:>AzCP:<4��>?<6M2@C@A�T

b

b + a+

a

a + c+

c

c + b≤ 1

1 + p+

2√

p

1 +√

p

f� CK: i 4

a

a + b+

b

b + c+

c

c + a= 3 −

(

b

b + a+

a

a + c+

c

c + b

) ¢� 4a894�4OA+GB6�A�ß � à�CK8�8*67ADCP8*�r4�;"C@ky6�:<;N=B:B2�T"CPk

a

a + b+

b

b + c+

c

c + a≥ 3 −

(

1

1 + p+

2√

p

1 +√

p

)

=p

1 + p+

2√

p

1 +√

p

f� G>?<8\¢>AD=�H3ID=M�B4 ßK6�àE¢BC�A�CP8�8*? � i C547:>A�AE=�H3ID=M�B4 ß � à f¡M4�A

x = b/a¢y = c/b

¢36M:<;z = a/c

f�� G34M:Fß � à�J<4 i =BLF4M81

1 + x+

1

1 + y+

1

1 + z≤ 1

1 + p+

2√

p

1 +√

p

f ß��_à� =�AD4�A+GB6�A

xyz = 16M:<;

x¢y¢z ∈ [1/p, p]

fF� =SH<ID=7�B4�ß � à+¢0C�AXCP8a8+? � i CK4M:>A�AD=H<ID=7�B4 ß��_à�ku=BIz672P2r8+? i Gx¢y¢B6M:<;

zf�BT A+G34O8�T3LNL�4\AEI�T�CP:�ß��_à*¢7� 4OLF6\T"6M898+?<L�4�A+GB6�A

z = max{x¢y¢z} f�� G34M:y¢8*CP: i 4

xyz = 16�:<;

z ≤ p¢�� 4aL�?<8+A�GB6\�B4

1 ≤ z ≤ p6M:<;

1/p ≤ xy ≤ 1f ¡M4\A

t =√

xyf�� G347:

t2z = 16�:<;

1/√

p ≤ t ≤ 1f � CK: i 4

x + y ≥ 2√

xy = 2t¢� 4 GB6\�B4

1

1 + x+

1

1 + y=

2 + x + y

1 + x + y + xy= 1 +

1 − t2

1 + x + y + t2

≤ 1 +1 − t2

1 + 2t + t2= 1 +

1 − t

1 + t=

2

1 + t

f ß � à� 47: i 4>¢

1

1 + x+

1

1 + y+

1

1 + z≤ 2

1 + t+

1

1 + z=

2

1 + t+

t2

1 + t2f

¡M4�Ah(t) =

2

1 + t+

t2

1 + t2f � CK: i 4

h′(t) =−2(1 − t)(1 − t3)

(1 + t)2(1 + t2)2¢<C�A�ku=>2@25=��a8

A+GB67AhCK8$;34 i ID476�8+CK:<¥�=B:

(0, 1]f �y=B:38-4M�B?r47:>AD2�T ¢>ku=BI

1/√

p ≤ t ≤ 1¢

h(t) ≤ h(1/√

p) =1

1 + p+

2√

p

1 +√

p

f� G>CP8$H3ID=M�B478�CP:<4��>?<6M2@C@A�TFß��_à�6�:<; i =BLNH32K4�AD4M8zA+G34aH3ID=3=>k�=>k�ßK6�à f

� � �

ßuJ�à � G>CP8�CK8�AEID4M6�AE4M;"8+CKL�CP2P6�I�2�T f0� G34�AEI�6�:38*Hr=B8+C@ADCK=B:(a, b, c, d) 7→ (b, a, d, c)CK:"A+G34�CP:<4��>?<6M2@C@A�T

a

a + b+

b

b + c+

c

c + d+

d

d + a≤ 1

1 + p+

3 3√

p

1 + 3√

p

ß � àTBCK472K;38zA+G34 4M�B?3C �>6M2K4M:>A�CP:<4��>?<6M2@C@A�T

b

b + a+

a

a + d+

d

d + c+

c

c + b≤ 1

1 + p+

3 3√

p

1 + 3√

p

f� CK: i 4

a

a + b+

b

b + c+

c

c + d+

d

d + a= 4 −

(

b

b + a+

a

a + d+

d

d + c+

c

c + b

) ¢� 4a894�4OA+GB6�A�ß � à�CK8�8*67ADCP8*�r4�;"C@ky6�:<;N=B:B2�T"CPk

a

a + b+

b

b + c+

c

c + d+

d

d + a≥ 4 −

(

p

p + 1+

3 3√

p

1 + 3√

p

)

=p

p + 1+

3

1 + 3√

p

f� G>?<8\¢>AD=�H3ID=M�B4 ß5J�àE¢BC�A�CK8�8+? � i CK4M:>A�AD=�H<ID=7�B4 ß � à f¡M4�A

x = b/a¢y = c/b

¢u = c/d

¢B6�:<;v = d/a

f�� G34M:Fß � à�J<4 i =BLF4M81

1 + x+

1

1 + y+

1

1 + u+

1

1 + v≤ 1

1 + p+

3 3√

p

1 + 3√

p

f ß � à� =�AD4XA+GB67A

xyuv = 16�:<;

x¢y¢u¢v ∈ [1/p, p]

f�� =�H<ID=7�B4�ß � à*¢�C@A�CK8z8*? � i C547:>A�AD=H<ID=7�B4 ß � à�ku=BIz672P2r8+? i Gx¢y¢u¢B6�:<;

vf¡M4�A

t =√

xy6�:<;

s =√

uvf �BTFA+G34"8�TBLFLF4�AEI�TFCK:�ß � à+¢3� 4�LF6\TQ6M898+?<L�4A+GB67A

t ≤ sf�� G34M:y¢r8*CP: i 4

ts = 1¢3� 4 8-4M4aA+GB6�A

t ≤ 1 ≤ sf � ?<IæA+G34MIDL�=BID4>¢ 8+CK: i 4

s2/u = v ≤ p¢�� 4aGB6��B4

s2/p ≤ u¢B6M:<;"A+G>?<8�¢

s2/p ≤ u ≤ pf� =���¢>ku=BI���ò�4�;

s¢

max

{

u + v∣

∣

∣ uv = s2 ¢ s2

p≤ u ≤ p

}

= max

{

u +s2

u

∣

∣

∣

s2

p≤ u ≤ p

}

= p +s2

p

f� G>?<8\¢

1

1 + u+

1

1 + v=

2 + u + v

1 + u + v + uv= 1 − s2 − 1

1 + u + v + s2

≤ 1 − s2 − 1

1 + p +s2

p+ s2

=2 + p +

s2

p

1 + p +s2

p+ s2

=p

s2 + p+

1

1 + p

f ßu�\à

� ���

Ú 8*CP:<¥OCK:<4M�B?<672PC�ADC5478$ßu�\à�6M:<;Fß � à+¢�� 4a¥74�A1

1 + x+

1

1 + y+

1

1 + u+

1

1 + v≤ 2s

1 + s+

p

s2 + p+

1

1 + p

f¡M4�A

g(s) =2s

1 + s+

p

s2 + p

f � CP: i 4g′(s) =

2(s − p)(s3 − p)

(s + 1)2(s2 + p)2¢�A+G>CK8

kK?<: i ADC5=B:ÉGB6M8"6Q25= i 6M2zLF6_ò�CPLN?<L 67As = 3

√p¢�� G>C i G�CP8 CK:�A+G34QCK:>AD4MI��>672

(1, p)f

�S4 GB6\�B4g(1) = −1 +

p

1 + p< 0

6�:<;g(p) = − 2

p + 1+

1

p + 1< 0 ` � G34M: i 4B¢

maxs∈[1,p]

g(s) = g(

3√

p) ¢B6M:<;"A+G347ID47ku=BID4>¢

1

1 + x+

1

1 + y+

1

1 + u+

1

1 + v≤ 2 3

√p

1 + 3√

p+

p3√

p2 + p+

1

1 + p

=3 3

√p

1 + 3√

p+

1

1 + p

f� G>CP8$H3ID=M�B478�ß � à�6M:<; i =BLNH32K4�AD4M8zA+G34aH3ID=3=>k�=>k�ß5J�à f

��� ���H�����U����Z �������DBIB�! [�O����CH<i57"�$E5'1��.Q�� S (��(#p���FZ(#&���SQ�FJ$9� (3(#�� �1LQ#)+q�$3�MQ���PU1RS &(]�&�1hP�� ��� �U8

,7 7I� [�f?9���J<E<K$ o S��� &g5'(#SU���1R��SUPU�.$KD�&XNeS 1R&B��&�$9����$E57"��98'&�('�ZP *'E]�12�.]K�����1��

�������u�Q 5������������� ���������?�m #"J$�&'$�(?)�*�+�-�t�;�(%0A7B)p���� "8=3$�; �J)�C:��HZ0Bv�)�"�(J0�=>-u$�n� �7B$�03)�(?=U0UCE:$�k�;�H�0U;�~6;�H<* }X;�+�"�03)D7'sp$�(J&#0UH6)D(?28G�C: �;�"�0@(iC:I�"�;�H<2J)��K�L?M

n ≥ 3� \�^�c,RUL?M

x1

�x2

�. . .

�xn N L�T [ VJSAM5S e�L�]5L�\�RX^�f�� N L�]�V�V%f#a%P�MJP�\?M

x1x2 · · · xn = 1� � [ ]

n = 3\�^�c

n = 4� T�] [ e�LuMJP�\?M

1

x21 + x1x2

+1

x22 + x2x3

+ · · · +1

x2n + xnx1

≥ n

2

�

{ $�7�G�=U03$�Hl+�-�=|4<)&6"%$�&'$�(�)?"D�&QV%S@^�_�MJP�L/V%f N VJM5S�M5f�M5S [ ^�V x1 =

√

a2

a1

�x2 =

√

a3

a2

�. . .

�xn =

√

a1

an

� MJP�L_�S�e�L�^�S@^�L � f�\�RASAM3� N L�a [ �,L�V

a1

a2 +√

a1a3

+a2

a3 +√

a2a4

+ · · · +an

a2 +√

ana2

≥ n

2

�d SU^�a`L √

a1a3 ≤ a1 + a3

2

�. . .

� √ana2 ≤ an + a2

2

� SAMOVJf�� a`L�VOM [ V8P [�Y MJP�\�Ma1

a1 + 2a2 + a3

+a2

a2 + 2a3 + a4

+ · · · +an

an + 2a1 + a2

≥ n

4

� �!MJP�L � \�f#a%P�� � d a%P Y \�] � � ^�L � f�\�R@S�MB� ��Y L�P�\�e�L

(a1 + · · · + an)2 ≤ [a1(a1 + 2a2 + a3) + · · · + an(an + 2a1 + a2)]

·(

a1

a1 + 2a2 + a3

+ · · · +an

an + 2a1 + a2

) �

� � �

� G>?<8\¢BC�Az8+? � i 4M8zAE=�89G3=���A+GB6�A4(a1 + · · · + an)2 ≥ n[a1(a1 + 2a2 + a3) + · · · + an(an + 2a1 + a2)]

f� G>CP8�CP:<4��>?<6M2@C@A�T�CP8�6�:�C5;34M:>ADC�A�T"ku=BI

n = 4f � =BI

n = 3¢>C�A�CP8

a21 + a2

2 + a23 ≥ a1a2 + a2a3 + a3a1

¢� G>C i G"CK8zAEI�?r4B¢3J34 i 6M?<8942(a2

1 + a22 + a2

3 − a1a2 − a2a3 − a3a1)

= (a1 − a2)2 + (a2 − a3)

2 + (a3 − a1)2 ≥ 0

f� G>CP8 i =BLFHB254\AE478�A+G34�H<ID=3=>k f�� �B?<672PC�A�TNG3=>25;B8z� G347:

xi = 1ku=BIz672P2

if

I�*�X)&���n = 3

^E&��E&�� ���X���������\ ��JN%<����ENlNd 3���K��"�"#�+D���$9"BPU1R&��� 3�3Q�1+p.$�=�1R&�(�)�B�!I.*'�12^E&��M��(�MS (')���1�12�)2P'�.QB �FJS �2��SU��(K�

���������Q 5�����������������������?�m #"J$�&'$�(?)�*�+�-�t�;�(%0A7B)p���� "8=3$�; �J)�C:��HZ0Bv�)�"�(J0�=>-u$�n� �7B$�03)�(?=U0UCE:$�k�;�H�0U;��K�L?M

0 < m < 1� \�^�c/RUL?M

a�b�c ∈

[√m, 1/

√m]

��� ] [ e�LuMJP�\?Ma3 + b3 + c3 + 3(1 + m)abc

ab(a + b) + bc(b + c) + ca(c + a)≥ 1 +

m

2

�

{ $�7�G�=U03$�Hl+�- �?$�)87 { 254Z7B$�(?+�)?"�z�C:9<;�-�(J0B*�)DCE�!��CQ� { j �K�L?M

f(x) = −2x2 +(

m2 +3m)

x+(

m3 −3m2 −2m+2)

� � [ ]x ≥ m

�f ′(x) = −4x + m2 + 3m ≤ −4m + m2 + 3m = m(m − 1) < 0

�Y PZS>a%P!SU��T�RAS>L�VOMJP�\?M

f(x)S@VQc�L�aJ]5L�\�V%S@^�_ �d f�T�T [ V8L�MJP�\�M

a = m�b ∈ [m, 1]

� \�^�cc = 1

�'W P�L�^m = a ≤ b ≤ c = 1\�^�c

2[a3 + b3 + c3 + 3(1 + m)abc]

− (m + 2)[a2b + ab2 + b2c + bc2 + c2a + ca2]

= 2[m3 + b3 + 1 + 3(1 + m)mb]

− (m + 2)[m2b + mb2 + b2 + b + m + m2]

= 2b3 − (m2 + 3m + 2)b2 + (−m3 + 4m2 + 5m − 2)b

+(m3 − 3m2 − 2m + 2)

= (1 − b)[−2b2 + (m2 + 3m)b + (m3 − 3m2 − 2m + 2)]

= (1 − b)f(b) ≥ (1 − b)f(1) = (1 − b)m(m − 1)2 ≥ 0�

Y PZS>a%P���SUL�RUc�Va3 + b3 + c3 + 3(1 + m)abc

a2b + ab2 + b2c + bc2 + c2a + ca2≥ 1 +

m

2

�

� ���

� ?<H<H<=B8-4�A+GB6�Aa¢b¢c ∈ [

√m, 1/

√m]f �QC@A+G3=>?BA�25=B8*8a=>kz¥747:<4MI�672PC�A�T ¢y� 4i 6M:F6M898+?<L�4OA+GB6�A

a ≤ b ≤ cf�� G347:m ≤ a

c≤ b

c≤ 1

f� 47: i 4>¢Bku=BI

(m′, a′, b′, c′) =(

a

c,

a

c,

b

c, 1) ¢�� 4 GB6\�B4

m ≤ m′ = a′ ≤ b′ ≤ c′ = 1¢

6�:<;"A+G347ID47ku=BID4>¢a′3 + b′3 + c′3 + 3(1 + m′)a′b′c′

a′2b′ + a′b′2 + b′2c′ + b′c′2 + c′2a′ + c′a′2 ≥ 1 +m′

2

¢� G>C i G i 6M:FJ<4X�aIDC�A�AE47:N6�8

a′3 + b′3 + c′3 + 3(1 + m′)a′b′c′

a′2b′ + a′b′2 + b′2c′ + b′c′2 + c′2a′ + c′a′2 − 1

≥ m′(

1

2− 3a′b′c′

a′2b′ + a′b′2 + b′2c′ + b′c′2 + c′2a′ + c′a′2

) f�BT"A+G34 g � 4�� � :<4��>?<6M2@C@A�Ty¢

a2b + ab2 + b2c + bc2 + c2a + ca2

6≥ 6

√a6b6c6 = abc

f� 47: i 4>¢

1

2− 3abc

a2b + ab2 + b2c + bc2 + c2a + ca2≥ 0

f� CK: i 4

a : b : c = a′ : b′ : c′ ¢�� 4aGB6\�B4a3 + b3 + c3 + 3abc

a2b + ab2 + b2c + bc2 + c2a + ca2− 1

=a′3 + b′3 + c′3 + 3a′b′c′

a′2b′ + a′b′2 + b′2c′ + b′c′2 + c′2a′ + c′a′2 − 1

≥ m′(

1

2− 3a′b′c′

a′2b′ + a′b′2 + b′2c′ + b′c′2 + c′2a′ + c′a′2

)

= m′(

1

2− 3abc

a2b + ab2 + b2c + bc2 + c2a + ca2

)

≥ m

(

1

2− 3abc

a2b + ab2 + b2c + bc2 + c2a + ca2

) ¢6�:<;"A+G347ID47ku=BID4>¢

a3 + b3 + c3 + 3(1 + m)abc

a2b + ab2 + b2c + bc2 + c2a + ca2≥ 1 +

m

2

f��� ��� ���������f ������!�R,a��59��IB�U"�$MNeS �2���3Q�1RSE"#�3QBPU*��1�(l"BP &BP+�5'(#SU���1R��SUPU�.$��.�.][� S (#$iN%<9$J5K"L��8�M�ED#I��H 3�u���ECH<@5K".$H571��.Q.� S ('�(Bp���FZ(3&���S+Q�FJ$M�U(#(3�� �1LQB)Rq�$���Q���P 1RS &�8�&�(��TPU*�H]�12�.]K�����1��lI�*��12\^E&��

&�� ���Z��(��S (�)���1�12�)�P'�����+QBP SU��(Z�.Q# �FJSUPRP����

� � �

á Ø ë © f 7 4MHB2K6 i 47L�47:>A f �D�M�����Üì � � � ¢ � ��� � ª ä � � � é�ã � � � � i¯_ç�� ° Mä-êKç5è ã �� J±�ç5ä+ã � � é7ê � �<ã�èyê��®0è>ç$¯Mã\ä�é+ç�� �c´�>âæç � è�7å � �3ê � è � ±� �<ã ° ã� �ç�â � � ° ã� �ç�â �rf

¡M4�AP (z)

J<4�6M:�T�:<=B:<� i =B:38+A�6�:>A i =BLNH32K4*òNL�=B:BC i Hr=>2 T3:<=BL�CK672 f � G3=�� A+GB6�AA+G34MID4�CP8�6 i =BLFHB254+ò :B?<L�J34MIw8*? i G�A+GB67A |w| ≤ 1

6�:<; |P (w)| ≥ 1f

s f � � å �BêKç � è � � ° ç�â �yã-å�¬ 3ê 7ç@åPå�ã � � � �>ã�è ����ä MèyâEã f¡M4�A

P (z) = zn +an−1zn−1 + · · ·+a1z+a0f � ?<H3Hr=B894B¢7AD=RA+G34 i =B:>AEI�6MI�Ty¢A+GB67A |P (z)| < 1

ku=BIa6M2@2 i =BLNH32K4*ò�:B?<LFJ<47I�8z8+? i G�A+GB67A |z| ≤ 1

f ��=B:38+C5;34MIQ(z) = a0zn + a1zn−1 + · · · + an−1z + 1

f�� =�AE4OA+GB6�AQ(z) = znP

(

1

z

) ku=BIz 6= 0

f�� G>?<8\¢>CPk |z| = 1¢�A+G34M: |Q(z)| =

∣

∣

∣P(

1

z

)∣

∣

∣< 1

f� A�ku=>2P2K=��a8zA+GB6�A 2π

∫

0

|Q(eit)| dt < 2πf�� =�� 4-�B47I-¢

∫ 2π

0

Q(eit) dt =

∫ 2π

0

(

a0eint + a1ei(n−1)t + · · · + an−1eit + 1)

dt = 2π¢

8*CP: i 4 2π∫

0

eikt dt =1

ik

(

e2kπi − 1)

= 0ku=BI�672P2

k = 1¢2¢. . .¢nf

� 47: i 4>¢2π =

∣

∣

∣

∣

2π∫

0

Q(eit) dt

∣

∣

∣

∣

≤2π∫

0

|Q(eit)| dt < 2π¢36 i =B:>AEI�6�;BC i ADC5=B: f

sæs f � � å �BêKç � è � � ¬<çKè � � � � é7ê � �<ã�èyê� � ��è�� � Mè � �� � � � è ��®�èBç(¯Mã�äæé*ç�ê� ���²MãMâ �<è � å � ����Mè � � âæç�ã\èyâEã ��� � � Mè�� � � �3ã_ç ��� �BçKè ��± � ��ç��yã�� � ��ê�yã ã ��çæê � ä f

¡M4�AP (z) = zn + an−1zn−1 + · · · + a1z + a0

f ¡M4�Aζ = eiπ/n

f�� G34M:ζn = −1

¢ζ2n = 1

¢B6M:<; |ζk| = 1ku=BI

k = 0¢1¢2¢. . .¢2n − 1

f�� =���¢2n−1∑

k=0

∣

∣P (ζk)∣

∣ ≥∣

∣

(

P (1) + P (ζ2) + · · · + P (ζ2n−2))

−(

P (ζ) + P (ζ3) + · · · + P (ζ2n−1))∣

∣

=

∣

∣

∣

∣

∣

n−1∑

k=0

(

P (ζ2k) − P (ζ2k+1))

∣

∣

∣

∣

∣

ß � à6�:<;

n−1∑

k=0

(

P (ζ2k) − P (ζ2k+1))

=

n−1∑

k=0

(

(ζ2k)n − (ζ2k+1)n)

+

n−1∑

j=1

aj

n−1∑

k=0

(

(ζ2k)j − (ζ2k+1)j) f ß��_à

� =�AD4 A+GB67A(ζ2k)n − (ζ2k+1)n = (ζn)2k − (ζn)2k+1 = 1 − (−1) = 2

¢6�:<;FG347: i 4>¢n−1∑

k=0

(

(ζ2k)n − (ζ2k+1)n)

= 2nf ß � à

� � �

£ :�A+G34a=�A+G34MIzGB6�:<; ¢Bku=BIz4M6 i Gj = 1

¢2¢. . .¢n − 1

¢�� 4aGB6\�B4n−1∑

k=0

(

(ζ2k)j − (ζ2k+1)j)

= (1 − ζj)

n−1∑

k=0

(ζ2j)k

= (1 − ζj)1 − (ζ2j)n

1 − ζ2j= 0

¢ ß � à8*CP: i 4

(ζ2j)n = (ζ2n)j = 1f

� ?rJ38EADC@AD?BADCP:<¥aß � à�6M:<;Fß � à�CK:>AD=Nß��_à�6�:<;"?<8*CP:<¥aß � à+¢�� 4OA+G347:�GB6\�B42n−1∑

k=0

∣

∣P (ζk)∣

∣ ≥ 2n¢

k5ID=BL � G>C i Ga� 4a;34�;>? i 4OA+GB67A�A+G347ID4�LN?<8EA�J34�8-=BLF4k¢M��C@A+G

0 ≤ k ≤ 2n − 1¢>ku=BI� G>C i G |P (ζk)| ≥ 1

¢ i =BLNH32K4�ADCP:<¥OA+G34�H<ID=3=>k f��� ���i���������J ��HA� 3 ?�������KD��i5�$K�!��(#pXAK��(#p.$9�#*#S (3&�8'&�('�JP *'�]�12�.]K�����1��EI.*'�12M^E&��>&�� ���J��(�

S (�)���1�12�)�P7�����Q#P S��(7�I�*�@]�12�#� �Kp.S���(T ��\D�&3QjQ�����j�9�3Q#)+*�m � �ZI.*'���12�F � 12��F�)���FX][���J&�(#&�� ����S ��� o �#P *G�����Q#P S��(#��+�&#PRQ�12��J&# 3�B��!�2*'�B^cP *3&BP

w)&�(T 3X)R*������(G���jPU*#&#P |w| = 1

&�(�� |P (w)| ≥ 1� o QBP�P *#S ��S �M(��#P

�.Q�1+]�1RS ��S (Bp�$[S (j��SU�^a� �9P *'@^@L� �A�+qB(��B^E(jNf&���S FgQ�F Nd�#�[Q��Q��M,[1RS (')S ][�UB�

��������� 5�����������������Z�?��� � a [ ]5]5L�a5M5L�c �������������Z�Z�Z������� #"%$�&'$�(�)�*!+�-���4�;�H�z��#G�H�Cxp0Uz�4 { 254<$�$�7';�=�=U;�254<)�*�=B$��o0 / j H �%0>;Z$���$�H�zm�XH�03v�)?"�(%0B=|-�C�o0

/ j Hm�Z0B=|-�C { 4�;�H�o0@C��4�0UH�;��K�L?M

a�b�c N L/MJP�L/MJP�]5L�L�VJS>c�L�V [Zg \�M`]�SU\�^�_�R>L � \�^�clR>L�M ha

�hb

�hc N L�MJP�L\�R�M5SAM5f#c�L�VOM [ MJP�L�VJS>c�L�V

a�b�c� ]5L�V%T#L�a5M5S�e�L?R�� ��� ] [ e�L�MJP�\�M

h2a

b2 + c2· h2

b

c2 + a2· h2

c

a2 + b2≤(

3

8

)3 �

� (%(?)�HX=U0U;�7A7|- =|4<)�(8;�k )�(?$�7�G�=U03$�H +�- sbrbq<0�&�& �?$�4�H�(?$�H�C�9�)�;�v�)�"8=B$�H�C ��EXC � { j ~�?$�)D7 { 254�73$�(�+�)�"�z�Cy9<;�-�(%03*�)�C �!��C1� { j ~!;�H<* sbr ��r { k )�)�H�w�C��;?7�=�+�$�k�k )87UCy=|4<)��)�=|4<)?"�7U;�H<*�(��&QV%S@^�_�MJP�L g|[ ]5��f�R@\

ha = b sin C� MJP�LyK�\ Y [Zg�d SU^�L�V � \�^�c�MJP�L � ��� �� ^�L � f�\�RASAM3� ��Y L [ N M5\�SU^

h2a

b2 + c2=

b2 sin2 C

b2 + c2=

sin2 B sin2 C

sin2 B + sin2 C

≤ sin2 B sin2 C

2 sin B sin C=

1

2sin B sin C

�\�^�c�VJSU�/S@R@\�]�R�� �

h2b

c2 + a2≤ 1

2sin C sin A

\�^�c h2c

a2 + b2≤ 1

2sin A sin B

�

� � �

��?32@ADCPH32 TBCP:<¥OA+G3478-4�6�:<;"?<8*CP:<¥OA+G34X� 472@25� ¤ :<=��a:"CK:<4M�B?<672PC�A�T

sin A sin B sin C ≤ 3√

3

8ß5894�4 � � ¢BH f � � � à+¢�� 4a=3J>A�6MCP:h2

a

b2 + c2· h2

b

c2 + a2· h2

c

a2 + b2≤ 1

8(sin A sin B sin C)2

≤ 1

8

(

3√

3

8

)2

=

(

3

8

)3 f� �>?<6M2@C@A�TFG3=>2K;38�CPky6M:<;F=B:B2 T�C@krA+G34RAEI�CK6M:<¥\254RCP8$4��>?3CP2P67AD4MI�672 f� ï_óæï �*ï�î �_ï7ö� � � £�f � =�A�AE47LF6 4\Az672 f ¢ ��ã � ±�ã�êuä�ç�â �è ã ��7å@çæêKç�ã\é ¢�4�ID=B:BCP:<¥74M:y¢ ��� � � f

��� ��� �����U����e ��\�E��AB�E�M?T�ED#I2$�"�&�(�.����L$H����$J5K"L��8�Nd��:E59 [D\��Nd 3CH:E53��De�'<�/.��".$i�#&�� &=�S p3QB�1R&�$gNf&.� �U��12)+&.$g"�]�&�S (B8��"# 3=3A9 7IT���K"BD���C��M:9� m�#$l5'(#SU���1R��SUPU�%� �Z"�&�1R&��������$G"�&�1R&��������$ o ���2(#S &&�(�� �H�1��LLpB�B��S (3&

( � �����Q#P S��(#� ) 8TNd�����! [D o ��I2�E�RDBD# �$Z�9�3Q#�(B$ =�1R&�(�)�h8��i53�ECH: ���M<�Nd�C!�M$C�p3Q��B�( o S (3*\A'*#SU�F ��S p�*Z"#)+*'�#��� $9/[S (3*jDB��(#p.$9/[SU�PU(#&�FX89���!�R,g��5���I��".$7NeS �2���3Q�1RS["#�3QBPU*��1�(Z"BP &BP+57(BS���1R��SP ��$?���.][� S (B$>N%<9$@5K"L��8'�E,K<�"[I�<�DB�"jA��9�M #Nd�".$E/#&�12�L&�q#LS�j�MS p�*g"B)R*��#��� $K�EPU*��(3��$E:'12��)�h8�M�ED#I��H 3�����ECH<@5K".$ 5'1��.Q�� S (��(#p���FZ(#&���SQ�FJ$G� (3(#�� �1LQ#)+q�$H�MQ���PU1RS &.8T�������������-���!�! #"�"�$J�'&�(�)R*��,#&��U���j/��12�3���$i����$X57"��98���<��iC�:��>�H K5�/7 [�'$@:'1R&�('�3G,[1R&.S 1RSUL$>� o 8MA� 3 ?�������MD��i5�$M�H��(BpeAK��(#p.$�#*#S (3&�8K�i� N;�3Nd�+D3<�"# [/K� m�B$K,[1R&�(�+&�(#S $9?[Q�pB����� &B��S &.89/7 [�X59D���CX�#NO5���I�?3$7�M�B���1R$7,���$�5K"L��87,���CH<�" '��I�"���<i57"�"#<E:9D#<i5�$7��P *'�(#��$M:'12��)�h8M,7 'IB 3�;?9���J<E<K$ o S��� &f57(BS���1R��SP ��$ED�&GNeS 1R&#��&.$M����$!5K"L��8o �+Cdk��E��<9$���P�QB�3�('P $9?�Q�(7?[Q�&�(G�HQ�&.k.*'��(Bpg57(BS���1R��SP �H� ��I��)+*3(������p��!&�('�H"#)S�(�)�L$��JQ�*#&�(B$9�XQB 3LS $�#*#S (3&�8'&�(��\P *'�]�12�.]K�����1��

� ����� � ������������������ ��!�"�

#%$&�')( *+&,'.-0/1*�&2$43�*�56( *�7�'.-0/8:9<; =?><@A8:BDC@�; =FEHGAIKJALC9:MON�=PCQEH@?R�CBDC@�; =FEHGTSVUW@FEHNPCYXHZ JAZ<[]\:MQ^`_<C@

acbDd�e ( *+&,'.-0/1*�&2$43+f bDd /g:>OEWhH9<; hHij8W9<; =P><@�GAIKJALC9:MON�=PCQEH@�GkR?lm><hW9:MT=?CTEW@FGTSnXHLC><oH>Op 9K[MTE:qrLCtsug<@FC]9WCT@F; N�vxwyZ UnZHz,MGkvOCQp p8:9<; =?><@FG{CTB|CT@F; =?;HIKJALC9:MON�=PCQEH@?R�CBDC@�; =F;�S4wyZ }+Z<[M]hW9:GT~�JAZ 8�ZO}�>:>:9:@?>]^Y~�UW@FEHNPCKXWZ JAZ:[]\<MT^y_:CT@

( *+&,'.-0/1*�&2$43�*�5�( *!7�'.-0/g:>OEWhH9<; hHij8W9<; =P><@�GAIKJ{LC]9:M]NF=?CTEW@FGkR?l ><hH9:MQ=PCQEH@�GTSn�<MT=?@F; N�vt[TEH@F@�_Ds�J�MQq<;:�rMOv; p8W9<; =P><@�G{CBDC@�; =F;�IYJ{LC]9:M]NF=?CTEW@FGkR�CTB|CT@F; =?;�SV�H\O; p ; ox��><hWi<~��CT�2�`; iT\:MB~�Z �4Z:wn@F><G�G�BDMh�~��hW9:@?C��4\<M]hWi<~W�yM]>:v];�[MQ=P>�~<�:_:@�EHGy�jGP; M:~<[]\:MQ^�hDwV>:9<; h