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 .

F - 12

1. (Mathletes Dec. 86) Solve for (American Babylonian)

( ) ( )

2. (Mathletes Dec. 86) If ( ) and ( ) and (A. B.),

then solve for when ( ( )) ( (

)).

3. (Mathletes Dec. 86) Solve for if

:

:

4. (DesCartes 1979) Prove (

) (

)

.

5. (DesCartes 1979) Prove .

6. (DesCartes 1979) Two curves are said to be orthogonal at a point of

intersection if the tangents at are perpendicular. It is known that

( )

and ( )

intersect at . Determine whether the

curves are orthogonal at .

7. (DesCartes 1979) Determine all other points of intersection of and

in question # 6.

8. (DesCartes 1979) Find the equation to tangent to ( ) at any one of

other intersection points in question # 7.

9. (1973 DesCartes) Prove

.

10. (1973 DesCartes) Given that the equation has non

zero roots and , evaluate the sum of all possible terms

where the subscripts and are distinct.

F- G - 12 - 1987

1. (Sen. M. 1972) If

where and then find

.

2. (Sen. M. 1972) If | | where and then:

(a) (b) (c) and

(d) ( ) (e) none of these

3. (Sen. M. 1972) Inscribed in a circle is a quadrilateral having sides of lengths

25, 39, 52 and 60 taken consecutively. Find the length of the diameter of

this circle.

4. (Sen. M. 1975) A man can commute either by train or by bus. If he goes to

work on train in morning, he comes home on bus in afternoon and if he

comes home on the train in afternoon, he took the bus in the morning.

During days the man rode the train 9 times and rode the bus 8 times in

the morning and 15 times in the afternoon. Find .

5. (Sen. M. 1975) A positive integer with digits (Base 10) is chosen at

random with each equally likely to be chosen. Find the probability that

.

6. (DesCartes 1986) Find

.

7. (DesCartes 1986) Write

in form .

8. (Sen. M. 1971) If is a non real root of then find the product

( )( ).

9. (Sen. M. 1972) If ( ) are complex roots of equation

where and then find in terms of and .

10. (Math 1983) Simplify ( ) ( ) .

F - X - 12

1. Find the remainder when is divided by 100 if .

2. Let be a triangle with . Find the area

of its circumcircle.

3. Without using your calculator, arrange √ √

√

in increasing

order.

4. In rectangle and . The angle bisector of

intersect at and the angle bisector of intersect at . and

intersect at . Find .

5. If

√

. If √ , where are integers and is

not divisible by any perfect squares other than 1, find .

6. If has positive divisors, how many positive divisors does

have?

7. Find .

8. What is the sum of the coefficients of the expanded form of

( ) ?

9. Let equal the sum of the 6 distinct prime factors of . What is ?

10. If three successive rolls of a die are all prime numbers, what is the

probability that they are all the same?

(a)

(b)

(c)

(d)

(e) none of these

11. Evaluate ( ( )).

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 .

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

F - G - 14

AHSME 1989

1. Five people are sitting at a round table. Let be the number of

people sitting next to at least one female and be the number of

people sitting next to at least one male. The number of possible ordered

pairs ( ) is

(a) 7 (b) 8 (c) 9 (d) 10 (e) 11

2. In a certain cross-country meet between two teams of five runners each,

a runner who finishes in the position contributes to his team’s score.

The team with the lower score wins. If there are no ties among the runners,

how many different winning scores are possible?

(a) 10 (b) 13 (c) 27 (d) 120 (e) 126

3. A regular octahedron is formed by joining the centers of adjoining faces

of a cube. The ratio of the volume of the octahedron to the volume of the

cube is

(a) √

(b)

√

(c)

(d)

√

(e)

4. Let be a positive integer. If the equation has 28

solutions in positive integers and , then must be either

(a) 14 or 15 (b) 15 or 16 (c) 16 or 17 (d) 17 or 18 (e) 18 or 19

5. Find the sum of the roots of that are between

and radians.

(a)

(b) (c)

(d) (e)

6. Find ∑ ( ) (

) , where (

)

( ) .

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

7. Suppose that 7 boys and 13 girls line up in a row. Let be the number of

places in the row where a boy and a girl are standing next to each other.

For example, for the row we have .

The average value of (if all possible orders of these 20 people are

considered) is closest to

(a) 9 (b) 10 (c) 11 (d) 12 (e) 13

8. Semicircle has center and radius 1. Point is on and

. Extend and to and , respectively, so that circular arcs

and have and as their respective centers. Circular arc has

center . The area of the shaded “smile”, , is

(a) ( √ ) (b) √ (c) ( √

)

(d)

√ (e) ( √ )

9. A circle of radius has chords of length 10 and of length 7. When

and are extended through and , respectively, they intersect at ,

which is outside the circle. If and then

(a) 70 (b) 71 (c) 72 (d) 73 (e) 74

10. Let √ . The product of the real parts of the roots of

is

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

(e)

A B C

D

E F

11. An “unfair” coin has a

probability of turning up heads. If this coin is

tossed 50 times, what is the probability that the total number of heads is

even?

(a) (

)

(b)

(

) (c)

(d)

(

)

(e)

12. Let be an isosceles trapezoid with bases and

. Suppose and a circle with center on is tangent

to segments and . If is the smallest possible value of , then

(a) 1369 (b) 1679 (c) 1748 (d) 2109 (e) 8825

13. A particle moves through the first quadrant as follows. During the first

minute it moves from the origin to ( ). Thereafter, it continues to follow

the directions indicated in the figure, going back and forth between the

positive and axes, moving one unit of distance parallel to an axis in

each minute. At which point will the particle be after exactly 1989 minutes?

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

(e) ( )

0

1

2

3

4

1 2 3

y

x