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1-st German Federal Mathematical Competition1970/71

First Round

1. The numbers 1, 2, . . . , 1970 are written on a board. It is allowed to erase

two numbers and write down their difference instead. This operation isrepeated until only one number remains. Prove that this number is odd.

2. We are given a piece of paper. We cut it into 8 or 12 arbitrary pieces,then we choose one of the obtained pieces and either cut it into 8 or 12pieces or do not cut it, etc. Can we obtain exactly 60 pieces in this way?Prove that every number of pieces greater than 60 can be obtained.

3. Suppose five segments are given such that any three of them are sides ofa triangle. Prove that some three of the segments are sides of an acute-angled triangle.

4. Let P and Q be two horizontally neighboring fields of an nn chessboard,P being to the left ofQ. A piece standing on field P is to be moved aroundthe chessboard. In each move, the piece can be moved to the neighboringfield to the right, up, or down-left. (For instance, it can be moved frome5 to e6, f5 or d4.) Prove that for no value ofn can the piece visit everyfield of the chessboard exactly once and end up on field Q.

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1-st German Federal Mathematical Competition1970/71

Second Round

1. Let a,b,c,d be natural numbers with ab = cd. Prove that a2 + b2 +c2 +d2

cannot be a prime number. Formulate and prove a generalization of thisstatement.

2. The inhabitants of a certain planet speak a language whose words consist

of letters A and O only. For the aim of avoidance of errors, every two

words of the same length differ in at least three positions. Prove that

there are no more than2n

n + 1words of length n.

3. Every two towns in a country are connected by a one-way road. Prove

that there is a town which can be reached from any other town eitherdirectly or passing through at most one other town.

4. A non-selfintersecting polygonal line of a length exceeding 1000 is given

inside a unit square. Show that for every such polygonal line, there is a

line parallel to one of the sides of the square that intersects the polygonal

line in at least 501 points.

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2-nd German Federal Mathematical Competition1971/72

First Round

1. In each cell of an n

n board there is a real number. Assume that the

sum of the numbers in each cross (formed by a row and a column) is atleast A. Find the smallest possible sum of all numbers in the board.

2. Let n 3 equal round beer coasters B1, B2, . . . , Bn be placed on a flattable so that Bi touches Bi+1 for i = 1, 2, . . . , n (where Bn+1 = B1).Another beer coaster B rolls along the exterior boundary of the chain ofcoasters. How many revolutions will coaster B make before reaching itsinitial position?

3. Suppose 2n1 subsets of an n-element set are selected in such a way thatany three of them have an element in common. Prove that all the subsetshave a common element.

4. Show that, when n goes through all natural numbers, the sequencen +

n + 1

2

contains all natural numbers except for the perfect squares.

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2-nd German Federal Mathematical Competition1971/72

Second Round

1. A knight is placed on a field of an infinite chessboard. Find the number

of fields the knight can reach in n moves.

2. Prove that among any 79 consecutive natural numbers there is one whosesum of digits is divisible by 13. Also show that this statement is not truefor 78 numbers.

3. The arithmetic mean of two distinct natural numbers x and y is a two-digit number. If the digits of this number are exchanged, the obtainednumber is equal to the geometric mean ofx and y.

(a) Determine x and y.

(b) Show that the solution in base g = 10 is unique, but that there areno solutions in base g = 12.

(c) Give further examples of bases g for which there exist such numbersx and y, and those for which there are no such numbers.

4. There are p participants in a chess tournament, and any two of them playat most one match against each other. After n matches are played, in each

group of three players there exist two who have not yet played against eachother. Prove that n p2/4.

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3-rd German Federal Mathematical Competition1972/73

First Round

1. A natural number has 1000 decimal digits, at most one of which is different

from 5. Prove that this number is not a perfect square.

2. From each of the points A and B on a flat lake one can reach any point on

the coast by a straightforward travel. Show that one can also reach every

point on the coast by a straightforward travel starting from any point on

the segment AB.

3. Let be given n arbitrary digits a1, . . . , an. Does there exist a natural

number such that the decimal digits of its square root after the decimal

point are exactly these n digits in that order? Justify your answer.

4. There are n people sitting at a round table. Suppose that the number

of persons whose neighbors to the right are of the same sex is the same

as the number of persons for which this does not hold. Prove that n is

divisible by 4.

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3-rd German Federal Mathematical Competition1972/73

Second Round

1. Let be given 51 points in a square of side 7. Prove that among these points

there always exist three that lie inside a circle with radius 1.

2. The following operations on a natural number are permitted:

(i) write digit 4 at the end of its decimal representation;

(ii) write digit 0 at the end of its decimal representation;

(iii) divide it by 2 if it is even.

Show that, starting with number 4, we can obtain every positive integer

using finitely many operations (i), (ii), (iii).

3. The floor of a rectangular room can be tiled with rectangular tiles 2 2and 4 1. Prove that if we replace one tile with a tile of the other type,then a tiling will no longer be possible.

4. Prove that for every positive integer n there is a natural number whosebase 10 representation consists only of digits 1 and 2, and that is divisibleby 2n. Is the statement true in bases 4 and 6?

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4-th German Federal Mathematical Competition1973/74

First Round

1. Find the necessary and sufficient condition that all convex quadrilaterals

formed out of a given four-bar linkage are trapezoids.

2. Seven polygons of area 1 lie in the interior of a square with side length 2.Show that there are two of these polygons whose intersection has an areaat least 1/7.

3. For an n-element set M, let P denote the family of all subsets ofM. Howmany pairs (A,B) of subsets from P are there such that A is a subset ofB?

4. All diagonals of a convex polygon are drawn. Prove that its sides anddiagonals can be assigned arrows in such a way that no round trip alongsides and diagonals is possible.

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Second Round

1. On a plane are given 25 points such that among any three of them, some

two are on a distance smaller than 1. Show that among the given pointsthere are 13 that can be covered by a disk of radius 1. Also prove thegeneralization of this statement.

2. There are 30 apparently equal balls, 15 of which have the weight a and theremaining 15 have the weight b, a = b. The balls are to be partitioned intotwo groups of 15, according to the weight. An assistant partitioned theminto two groups, and we wish to check if his partition is correct. How canwe check that with as few weighings as possible?

3. A circle K1 of radius 1/2 is inscribed in a semi-circle H with diameterAB and radius 1. A sequence of different circles K1, K2, . . . with radiir1, r2, . . . respectively are drawn so that, for each n 1, circle Kn+1 istangent to H, Kn, and AB. Prove that an = 1/rn is an integer for eachn N, and that it is a perfect square for n even and two times a perfectsquare for n odd.

4. Peter and Paul gamble as follows. For each natural number, successively,they determine its largest odd divisor and compute its remainder when

divided by 4. If this remainder is 1, then Peter pays Paul 1 DM (Deutschemark); otherwise Paul pays Peter 1 DM. After some time they stop playingand balance the accounts. Prove that Paul wins.

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First Round

1. A game is played with two heaps ofp and q stones. Two players alternate

playing, with A starting. A player in turn takes away one heap and dividesthe other heap into two smaller ones. A player who cannot perform a legalmove loses the game. For which values ofp and q can A force a victory?

2. A line g and a polint A outside g are given in a plane. A point P movesalong g. Find the locus of the third vertices of equilateral triangles whosetwo vertices are A and P.

3. A natural number n is called breakable if there exist positive integers

a,b,x,y such that a + b = n and

x

a +

y

b = 1. Find all breakable numbers.

4. A number of unit discs are given inside a square of side 100 such that

(i) no two of the discs have a common interior point, and

(ii) every segment of length 10, lying entirely within the square, meetsat least one disc.

Prove that there are at least 400 discs in the square.

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Second Round

1. Starting at (1, 1), a stone is moved in the coordinate plane according to

the following rules:

(i) From any point (a, b), the stone can move to (2a, b) or (a, 2b).

(ii) From any point (a, b), the stone can move to (a b, b) if a > b, or to(a, b a) if a < b.

For which positive integers x, y can the stone be moved to (x, y)?

2. Let S be a union of finitely many disjoint subintervals of [0, 1] such thatno two points in S have distance 1/10. Show that the total length of theintervals comprising S is at most 1/2.

3. Each diagonal of a convex pentagon is parallel to one side of the pentagon.Prove that the ratio of the length of a diagonal to that of its correspondingside is the same for all five diagonals, and compute this ratio.

4. Prove that every integer k > 1 has a multiple less than k4 whose decimalexpension has at most four distinct digits.

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First Round

1. Can a square of side length 5 be covered by three squares of side length

4?

2. The cells of an nn board are labelled with the numbers 1 through n2 in

the usual way. Let n of these cells be selected, no two of which are in the

same row or column. Find all possible values of the sum of their labels.

3. Four lines are given in a plane so that any three of them determine a tri-

angle. One of these lines is parallel to a median in the triangle determined

by the other three lines. Prove that each of the other three lines also has

this property.

4. Find all positive integers n for which n 2n1 + 1 is a perfect square.

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Second Round

1. For a given set of points in space it is allowed to mirror a point from the

set with respect to another point from the set, and to include the imagein the set. Starting with a set of seven vertices of a cube, is it possible toinclude the eight vertex in the set after finitely many such steps?

2. The sequence z0, z1, z2, . . . is defined by z0 = 0 and

zn =

zn1 +3r 1

2ifn = 3r1(3k + 1) for some integers r,k;

zn1 3r + 1

2ifn = 3r1(3k + 2) for some integers r,k.

Prove that every integer occurs exactly once in this sequence.

3. Rectangles ABB1A1, BCC1A2, CAA2C2 are constructed in the exteriorof a triangle ABC. Prove that the perpendicular bisectors of the segmentsA1A2, B1B2, and C1C2 are concurrent.

4. Let p be an odd prime number. Find the positive integers x and y withx y for which 2pxy is the smallest possible.

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First Round

1. Given 100 integers, is it always possible to choose 15 of them such that

the difference of any two of the chosen numbers is divisible by 7? Whatis the answer if 15 is replaced by 16?

2. Determine all prime numbers p for which the system of equations

p + 1 = 2x2

p2 + 1 = 2y2

has a solution in integers x, y.

3. A square Sa is inscribed in an acute-angled triangle ABCwith two verticeson side BC and one on each of sides AC and AB. Squares Sb and Sc are

analogously inscribed in the triangle. For which triangles are the squares

Sa, Sb, and Sc congruent?

4. There are 10000 trees in a park, arranged in a square grid with 100 rows

and 100 columns. Find the largest number of trees that can be cut down,

so that sitting on any of the tree stumps one cannot see any other tree

stump.

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Second Round

1. Three faces of a regular tetrahedron are painted in white and the remaining

one in black. Initially, the tetrahedron is positioned on a plane with theblack face down. It is then tilted several times over its edges. After a whileit returns to its original position. Can it now have a white face down?

2. Show that for any rational number a the equation y =x2 + a has in-

finitely many solutions in rational numbers x and y.

3. A semicircle with diameter AB = 2r is divided into two sectors by anarbitrary radius. To each of the sectors a circle is inscribed. These twocircles touch AB at S and T. Show that ST

2r(

2

1).

4. Prove that ifn is a natural number such that both 3n+ 1 and 4n+ 1 aresquares, then n is divisible by 56.

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Second Round

1. Find all integer solutions (x,y,z) of the equation xy + yz + zxxyz = 2.

2. Prove that there exist 16 subsets of set M = {1, 2, . . . ,10000} with thefollowing property: For every z M there are eight of these subsets whoseintersection is {z}.

3. A triangle ABC satisfies BC = AC+ 12AB. Point P on side AB is taken

so that AP = 3PB. Prove that PAC= 2CPA.

4. Let 3(2n 1) points be selected in the interior of a polyhedron P withvolume 2n, where n is a positive integer. Prove that there exists a convex

polyhedron U with volume 1, contained entirely inside P, which containsnone of the selected points.

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First Round

1. 1600 coconuts are distributed over 100 monkeys, allowing some monkeys

to be left empty-handed. Prove that, independent of the distribution,there always exist four monkeys with the same number of coconuts.

2. The sequences (an) and (bn) are defined by a1 = b1 = 1 and

an+1 = an + bn, bn+1 = anbn for n = 1, 2, . . .

Show that every two distinct terms of the sequence (an) are coprime.

3. In the plane are given a segment AC and a point B on the segment. Let us

draw the positively oriented isosceles triangles ABS1, BC S2, and CAS3with the angles at S1, S2, S3 equal to 120

. Prove that the triangle S1S2S3is equilateral.

4. It is known that there are polyhedrons whose faces are more numberedthan the vertices. Find the smallest number of triangular faces that sucha polyhedron can have.

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Grades 11-13

First Day

1. Prove that there are no perfect squares a,b,c such that ab bc = a.

2. For a positive integer k, let us denote by u(k) the greatest odd divisor ofk. Prove that, for each n N,

1

2n

k = 12

n u(k)

k>

2

3.

3. In a convex quadrilateral ABCD we are given that

CB D = 10, CAD = 20, ABD = 40, BAC = 50.

Determine the angles BC D and ADC.

Second Day

4. Find all real solutions (x,y,z) of the system of equations

x3 = 2y 1,y3 = 2z 1,z3 = 2x 1.

5. We are given n discs in a plane, possibly overlapping, whose union hasthe area 1. Prove that we can choose some of them which are mutually

disjoint and have the total area greater than 1/9.

6A. Let us define f and g by

f(x) = x5 + 5x4 + 5x3 + 5x2 + 1,g(x) = x5 + 5x4 + 3x3 5x2 1.

Determine all prime numbers p such that, for at least one integer x, 0 x < p 1, both f(x) and g(x) are divisible by p. For each such p, find allx with this property.

6B. An approximate construction of a regular pentagon goes as follows. In-scribe an arbitrary convex pentagon P1P2P3P4P5 in a circle. Now choosean arror bound > 0 and apply the following procedure.

(a) Denote P0 = P5 and P6 = P1 and construct the midpoint Qi of thecircular arc Pi1Pi+1 containing Pi.

(b) Rename the vertices Q1, . . . , Q5 as P1, . . . , P 5.

(c) Repeat this procedure until the difference between the lengths of thelongest and the shortest among the arcs PiPi+1 is less than .

Prove this procedure must end in a finite time for any choice of and thepoints Pi.

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germaniabw 1970-1999

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