D - E - G - 12

1. (Mathletes Dec. 1986) Solve for , .

2. (Mathletes Dec. 1986) Solve for , .

3. (Mathletes Dec. 1986) Solve for , √ √ √ .

4. (Mathletes Dec. 1986) Solve for and ,

{

5. (Mathletes Dec. 1986) Solve for ,

.

6. (Freshman) If then find .

7. (Freshman) If and then find in terms of and .

8. (Freshman) If the roots of are and , then find the value

of

.

9. (Euclid 1981) Find the next integer larger than (√ √ ) .

10. (Euclid 1980) If √

√ , find .

D - E - F - G - 12

1. (Mathletes 1986) Solve for ,

( ) ( ) .

2. (Mathletes 1986) Solve for ( ),

and

3. (Mathletes 1986) Solve for , .

4. (Mathletes 1986) Solve for and if

and

5. (Mathletes 1986) Solve for ,

√ ( )

.

6. (Mathletes 1986) Solve for ,

| | | | .

7. (Euclid 1980) Find the area of the triangle whose vertices are: point ( ),

the vertex of and midpoint of the line segment

determined by points of intersection of the line with

.

8. (DesCartes 1969) is an acute angled triangle with circumcentre .

Prove that if is altitude to , then bisector of bisects .

9. (Math. 1965) If and they are not equal and their sum is zero and

their product is , then find .

10. (Math. 1966) and are points on and respectively. If ( )

is midpoint of and | | , then find the equation (locus) of .

D - E - F - G - 13

1. (1966 Jr. - adapt) Find the least positive integer having the remainders 1, 4

and 1 when divided by 3, 5 and 11.

2. (1966 Jr. - adapt) and are points on the lines and

respectively. If ( ) is midpoint of and length of then find the

equation of locus of .

3. (1985 Sen. Math) A non-zero digit is chosen in such a way that the

probability of choosing digit is ( ) ( ). The probability that

digit is chosen is exactly

the probability that the digit chosen is in set

(a) { } (b) { } (c) { } (d) { }

(e) { }

4. (1981 Sen. Math) Find the number of real solutions to the equation

.

5. (1981 Sen. Math) If and are positive numbers and then the

number obtained by increasing by and decreasing the result by

exceeds (if and only if)

(a) (b)

(c)

(d)

(e)

6. (1982 Sen. Math) Let [ ] denote the greatest integer not exceeding . Let

satisfy { [ ]

[ ] . If is not an integer then is

(a) an integer (b) between and (c) between and

(d) between and (e)

(graph)

7. (1972 DesCartes) Lines and meet at so thet

and the angle is . A circle is drawn to pass through

. Determine the radius of this circle.

8. (1977 DesCartes) A point is chosen inside an irregular convex pentagon

. Prove

( ( )) ( ( )) ( ( )) ( ( )) ( ( ))

( ( )) ( ( )) ( ( )) ( ( )) ( ( )).

9. (1980 DesCartes) Calculate the coordinates of the foot of the perpendicular

from ( ) to line .

10. (1980 DesCartes) If , evaluate .

D - E - G - 13

1. (1966 Jr.) The expressions and ( )( ) are

(a) always equal

(b) never equal

(c) equal if

(d) equal if

(e) equal only if

(Note wording - ask )

2. (1966 Jr.) A rectangular sheet of paper is 6 cm wide and 8 cm in

length. The paper is folded so that the two diagonally opposite corners

coincide. Find the length of the crease in the paper in cm.

3. (1965 Jr. - adapt) A man walks from to at 4 km/hr and from to at

3 km/hr. Then he walks from to at 6 km/hr and from to at 4 km/hr.

If the total time taken is 6 hours and then find the total distance

walked.

4. (1985 Sen. Math) Six bags of marbles contain 18, 19, 21, 23, 25, and 34

marbles, respectively. One bag contains only chipped marbles. The other

5 bags contain no chipped marbles. Jane takes 3 bags and George 2 and

only the chipped marbles bag remains. If Jane gets twice as many as

George, how many chipped marbles are there?

5. (1981 Sen. Math) If and ( ) ( ) then find .

6. (1981 Sen. Math) In the adjoining figure, is a diagonal of the cube. If

has length then find the surface area of the cube.

P

Q

7. (1981 Sen. Math) The three sides of a right triangle have integral lengths

which form an arithmetic progression. One side could be

(a) 22 (b) 58 (c) 81 (d) 91 (e) 361

8. (1976 Euclid) A circle has an inscribed triangle whose sides are √ √

and . Find the measure in degrees of the angle subtended at the centre

of the circle by the shortest side.

9. (1978 Euclid) Prove that is divisible by positive integral

values of .

10. (1978 Euclid) The straight line is reflected in the line

. Find the equation of its image.

D - E - F - G - 13

1. (1966 Jr. - adapt) Find the least positive integer having the remainders 1, 4

and 1 when divided by 3, 5 and 11.

2. (1966 Jr. - adapt) and are points on the lines and

respectively. If ( ) is midpoint of and length of then find the

equation of locus of .

3. (1985 Sen. Math) A non-zero digit is chosen in such a way that the

probability of choosing digit is ( ) ( ). The probability that

digit is chosen is exactly

the probability that the digit chosen is in set

(a) { } (b) { } (c) { } (d) { }

(e) { }

4. (1981 Sen. Math) Find the number of real solutions to the equation

.

5. (1981 Sen. Math) If and are positive numbers and then the

number obtained by increasing by and decreasing the result by

exceeds (if and only if)

(a) (b)

(c)

(d)

(e)

6. (1982 Sen. Math) Let [ ] denote the greatest integer not exceeding . Let

satisfy { [ ]

[ ] . If is not an integer then is

(a) an integer (b) between and (c) between and

(d) between and (e)

(graph)

7. (1972 DesCartes) Lines and meet at so thet

and the angle is . A circle is drawn to pass through

. Determine the radius of this circle.

8. (1977 DesCartes) A point is chosen inside an irregular convex pentagon

. Prove

( ( )) ( ( )) ( ( )) ( ( )) ( ( ))

( ( )) ( ( )) ( ( )) ( ( )) ( ( )).

9. (1980 DesCartes) Calculate the coordinates of the foot of the perpendicular

from ( ) to line .

10. (1980 DesCartes) If , evaluate .

D - E - G - 13

1. (1966 Jr.) The expressions and ( )( ) are

(a) always equal

(b) never equal

(c) equal if

(d) equal if

(e) equal only if

(Note wording - ask )

2. (1966 Jr.) A rectangular sheet of paper is 6 cm wide and 8 cm in

length. The paper is folded so that the two diagonally opposite corners

coincide. Find the length of the crease in the paper in cm.

3. (1965 Jr. - adapt) A man walks from to at 4 km/hr and from to at

3 km/hr. Then he walks from to at 6 km/hr and from to at 4 km/hr.

If the total time taken is 6 hours and then find the total distance

walked.

4. (1985 Sen. Math) Six bags of marbles contain 18, 19, 21, 23, 25, and 34

marbles, respectively. One bag contains only chipped marbles. The other

5 bags contain no chipped marbles. Jane takes 3 bags and George 2 and

only the chipped marbles bag remains. If Jane gets twice as many as

George, how many chipped marbles are there?

5. (1981 Sen. Math) If and ( ) ( ) then find .

6. (1981 Sen. Math) In the adjoining figure, is a diagonal of the cube. If

has length then find the surface area of the cube.

P

Q

7. (1981 Sen. Math) The three sides of a right triangle have integral lengths

which form an arithmetic progression. One side could be

(a) 22 (b) 58 (c) 81 (d) 91 (e) 361

8. (1976 Euclid) A circle has an inscribed triangle whose sides are √ √

and . Find the measure in degrees of the angle subtended at the centre

of the circle by the shortest side.

9. (1978 Euclid) Prove that is divisible by positive integral

values of .

10. (1978 Euclid) The straight line is reflected in the line

. Find the equation of its image.

F - G - 13

1. (DesCartes Warm-up) The graph of the function , and

constants to be determined, passes through the points ( ) (

√ ) and

(

√ ). Find , and .

2. (DesCartes Warm-up) The graph of the function passing

through point ( ) has a maximum value of at

. Find , and .

3. (Sen. Math 1985) If √

and

√

where , then which of the

following is not correct (show why).

(a) (b) (c)

(d) (e)

4. (Sen. Math 1985) If and , then find the value of

( )( ).

5. (Sen. Math 1985) How many distinguishable rearrangements of the letters in

CONTEST have both vowels first?

6. (Sen. Math 1984) The value of

is:

(a) (b)

(c)

( ) (d) (e)

7. (Sen. Math 1981) Alice, Bob and Carol repeatedly take turns tossing a die. Alice

begins, Bob always follows Alice, Carol always follows Bob and Alice always follows

Carol. Find the probability that Alice will be the first to roll a six.

8. (DesCartes 1971) A bi-tangent is a line tangent to a curve at two distinct points. Find

the bi-tangent to .

9. (DesCartes 1975) Given cubic equation with rational coefficients

has the root √ , determine the values of and .

10. (DesCartes 1977) If is an acute angle such that

, find and

.

D - E - F - G - 14

AHSME 1997

1. Which one of the following integers can be expressed as the sum of 100

consecutive positive integers?

(a) 1,627,384,950 (b) 2,345,678,910 (c) 3,579,111,300

(d) 4,692,581,470 (e) 5,815,937,260

2. For any positive integer , let ( ) {

What is ∑ ( ) ?

(a) (b) (c)

(d)

(e)

3. Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole

number of dollars to spend, and together they had $56. The absolute

difference between the amounts Ashley and Betty had to spend was $19.

The absolute difference between the amounts Betty and Carlos had was

$7, between Carlos and Dick was $5, between Dick and Elgin was $4, and

between Elgin and Ashley was $11. How much did Elgin have?

(a) $6 (b) $7 (c) $8 (d) $9 (e) $10

4. In the figure, polygons and are isosceles right triangles; and

are squares with sides of length 1; and is an equilateral triangle. The

figure can be folded along its edges to form a polyhedron having the

polygons as faces. The volume of this polyhedron is

(a)

(b)

(c)

(d)

(e)

A

B

C D E

F G

5. A rising number, such as 34689, is a positive integer each digit of which

is larger than each of the digits to its left. There are ( ) five-digit

rising numbers. When these numbers are arranged from smallest to

largest, the 97th number in the list does not contain the digit

(a) 4 (b) 5 (c) 6 (d) 7 (e) 8

6. Let be a parallelogram and let ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗ ⃗⃗⃗⃗ ⃗⃗ and ⃗⃗⃗⃗ ⃗⃗ ⃗ be parallel

rays in space on the same side of the plane determined by . If

and and are the midpoints of

and , respectively, then

(a) 0 (b) 1 (c) 2 (d) 3 (e) 4

7. Triangle and point in the same plane are given. Point is

equidistant from and , angle is twice angle , and intersects

at point . If and then

(a) 5 (b) 6 (c) 7 (d) 8 (e) 9

8. Consider those functions that satisfy ( ) ( ) ( ) for all

real . Any such function is periodic, and there is a least common positive

period for all of them. Find .

(a) 8 (b) 12 (c) 16 (d) 24 (e) 32

A B

C

P

D

9. How many ordered triples of integers ( ) satisfy

| | and | | ?

(a) 0 (b) 4 (c) 6 (d) 10 (e) 12

10. Call a positive real number special if it has a decimal representation

that consists entirely of digits and . For example,

and are special numbers. What is the smallest such

that can be written as a sum of special numbers?

(a) 7 (b) 8 (c) 9 (d) 10

(e) cannot be represented as a sum of finitely many special numbers

11. For positive integers , denote by ( ) the number of pairs of different

adjacent digits in the binary (base two) representation of . For example,

( ) ( ) ( ) ( ) , and

( ) ( ) . For how many positive integers less than or

equal to does ( ) ?

(a) 16 (b) 20 (c) 26 (d) 30 (e) 35

12. A square flag has a red cross of uniform width with a blue square in the

center on a white background as shown. (The cross is symmetric with

respect to each of the diagonals of the square.) If the entire cross (both the

red arms and the blue center) takes up 36% of the area of the flag, what

percent of the area of the flag is blue?

(a) 0.5 (b) 1 (c) 2 (d) 3 (e) 6

RED RED

RED RED

BLUE

D - E - G - 14

AHSME 1997

1. If and are digits for which

then

(a) 3 (b) 4 (c) 7 (d) 9 (e) 12

2. The adjacent sides of the decagon shown meet at right angles. What is

its perimeter?

(a) 22 (b) 32 (c) 34 (d) 44 (e) 55

3. If and are real numbers such that

( ) ( ) ( )

then

(a) −12 (b) 0 (c) 8 (d) 12 (e) 50

4. If is larger than , and is larger than , then is what

percent larger than ?

(a) (b) (c) (d) (e)

2

8

12

2

5. A rectangle with perimeter 176 is divided into five congruent rectangles

as shown in the diagram. What is the perimeter of one of the five congruent

rectangles?

(a) 35.2 (b) 76 (c) 80 (d) 84 (e) 86

6. Consider the sequence whose nth term is

( ) . What is the average of the first 200 terms of the sequence?

(a) (b) (c) (d) (e)

7. The sum of seven integers is −1. What is the maximum number of the

seven integers that can be larger than 13?

(a) 1 (b) 4 (c) 5 (d) 6 (e) 7

8. Mientka Publishing Company prices its best seller Where’s Walter? as

follows:

( ) {

where is the number of books ordered, and ( ) is the cost in dollars of

books. Notice that 25 books cost less than 24 books. For how many values

of is it cheaper to buy more than books than to buy exactly books?

(a) 3 (b) 4 (c) 5 (d) 6 (e) 8

9. In the figure, is a 2 x 2 square, is the midpoint of , and is

on . If is perpendicular to , then the area of quadrilateral is

(a) 2 (b) √

(c)

(d) √ (e)

10. Two six-sided dice are fair in the sense that each face is equally likely

to turn up. However, one of the dice has the 4 replaced by 3 and the other

die has the 3 replaced by 4. When these dice are rolled, what is the

probability that the sum is an odd number?

(a)

(b)

(c)

(d)

(e)

11. In the sixth, seventh, eighth and ninth basketball games of the season,

a player scored 23, 14, 11, and 20 points, respectively. Her points-per-

game average was higher after nine games than it was after the first five

games. If her average after ten games was greater than 18, what is the

least number of points she could have scored in the tenth game?

(a) 26 (b) 27 (c) 28 (d) 29 (e) 30

12. If and are real numbers and , then the line whose equation

is cannot contain the point

(a) ( ) (b) ( ) (c) ( )

(d) ( ) (e) ( )

A

B C

D E

F

13. How many two-digit positive integers have the property that the sum

of and the number obtained by reversing the order of the digits of is a

perfect square?

(a) 4 (b) 5 (c) 6 (d) 7 (e) 8

14.

(a) (b) (c) (d) (e)

15. In and is on with . Find the

ratio .

(a) (b) (c) (d) (e)

A

B

C D