IN

MATHEMATICS

- Fill out completely the following information about yourself.
PRINT __________________________________________________________________________________________

Last name First name Initial Phone No.ADDRESS ________________________________________________________________________________________

Street address City State Zip

Your hight school: Name ____________________________________ City ____________________________

High School Counselor or Advisor: ______________________________________________________________ - This examination consists of two parts. The time allowed for each
will be approximately 60 minutes. Should you finish Part I early, you
may proceed to Part II.
- Part I consists of 16 objective-type questions. Each question has
five possible answers marked: A., B., C., D., E. Only
__one__Example: If

`x=5`and`y=-2`, then`x+4y`is(A)

`-3`(B)`-2`(C)`-1`(D)`0`(E)`+1`. - Part II consists of 4 subjective-type questions. Show a summary of
your work in this booklet for each question you attempt, whether or not
you obtain a complete solution. Scratch paper is provided
be sure to show the essential steps of your work concisely in the space provided for each question.*but*the work appearing in this booklet will be scored. You will be scored on your method of attack, ingenuity, insight, inventiveness, and logical developments as well as your solutions.*Only* - Pencils and scratch paper will be provided.
**No tables, rulers, compasses, protractors, slide rules, calculators, or other aids are permitted.** - The scoring of questions in Part I has been devised to
discourage random guessing and will be computed as follows:
`(three times number correct) - (number wrong)`. - The scoring for the three questions in Part II will be 13 points per question. Partial credit will be given so it will be to your advantage to do as much as you are able to do on each question.

- The scoring of questions in Part I has been devised to
discourage random guessing and will be computed as follows:
- For the scoring committee.
*Do not write in this space.***Part I:**

No. Correct: __________

No. Wrong: __________

**Part II:**

Score in 1: __________

Score in 2: __________

Score in 3: __________

Score in 4: __________

**Score in Part I:**__________

**Score in Part II:**__________

**T O T A L :**__________

- Let
`S`be the set of integers`0, 1, 2, . . . , 24, 25`. The number of members of*S*that leave a remainder of 0 when divided by 6 is:(A) (B)

(C)

(D)

(E)

4 5

6

7

8

- The vertex of the parabola
*y=3x*will be a point on the x axis if the value of^{2}+12x+b*b*is(A) (B)

(C)

(D)

(E)

36 12

-16

-12

-36

- Let
*a,b > 0.*When simplified, square root of*-(a*equals^{2}-b^{2})^{2}+(a^{2}+b^{2})^{2}(A) (B)

(C)

(D)

(E)

0 `ab`Square rot of

`a`^{2}+b^{2}`2ab`Square root of

`a`^{2}-b^{2} - The average of a set of 50 numbers is 38. If two numbers of the set,
namely 45 and 55, are discarded, the average of the remaining numbers is
(A) (B)

(C)

(D)

(E)

38.5 37.5

37

36.5

36

- When the base of a triangle is increased 20% and the altitude to this base
is decreased 20% the change in area is
(A) (B)

(C)

(D)

(E)

2% increase 4% increase

0%

2% decrease

4% decrease

`2`is equal to^{-(2k+1)}-2^{-(2k-1)}+2^{-2k}(A) (B)

(C)

(D)

(E)

`2`^{-2k}`2`^{-(2k-1)}`-2`^{-(2k+1)}`0``2`- If the line
`y=mx+1`intersects the ellipse`x`exactly once, then the value of^{2}+4y^{2}=1`m`is^{2}(A) (B)

(C)

(D)

(E)

__1__

2

__2__

3

__3__

4

__4__

5__5__

6

- If
`a`and`b`are real numbers the equation`3x-5+a=bx+1`has a unique solution`x`:(A) (B)

(C)

(D)

(E)

for all `a`and`b`if

`a not= 2b`if

`a`not= 6if

`b`not= 0if

`b`not= 3 - A triangle has angles of
`30`and^{o}`45`. If the side opposite the^{o}`45`angle has length 8, then the side opposite the^{o}`30`angle has length^{o}(A) (B)

(C)

(D)

(E)

4 4(square of 2)

4(square of 3)

4(square of 6)

6

- The number of cubic feet in the volume of a cube is the same as the number
of square inches in its surface area. The length of the edge, expressed in
feet, is
(A) (B)

(C)

(D)

(E)

6 864

1728

6 times 1728

2304

- The square root of
`-64`is(A) (B)

(C)

(D)

(E)

`64i``-64i``-8``8i`8

- The ratio of females to males in a math club is 7 to 4. If three females
and twelve males are absent from a meeting, the ratio of females to males
is 5 to 2. How many members of the math club attend the meeting?
(A) (B)

(C)

(D)

(E)

24 60

63

84

99

- Two cylists, 30 miles apart and starting at the same time, would be
together in 6 hours if they traveled in the same direction, but would
pass each other in 2 hours if they traveled towards each other in opposite
directions. The ratio of speed of the faster cylist to that of the slower
is:
(A) (B)

(C)

(D)

(E)

__2__

1

__3__

2

__2__

3

__4__

3

__3__

1 - In a general triangle
`ADE`(as shown) lines`EB`and`EC`are drawn. Which of the following angle relations is true?(A) (B)

(C)

(D)

(E)

`x+z=a+b``y+z=a+b``m+x=w+n``x+z+n=w+c+n``x+y+n=a+b+m` - Let
`S`be the sum of the first nine terms of the sequence`x+a, x`Then^{2}+2a, x^{3}+3a, . . .`S`equals:(A) __50 a+x+x__^{8}

`x+1`(B) `50 a-`__x+x__^{16}

`x-1`^{}(C) `x`^{4}-1^{ }

x+1

+ 45a(D) `x`^{10}-x^{ }

x-1`+ 45a`(E) `x`^{11}-x^{ }

x-1`+ 45a` - The derivative of
`x`^{3}+x^{2}`is`(A) (B)

(C)

(D)

(E)

`x`^{3}`3x`^{2}+2x`3x`^{2}`2x``3x`^{2}+2/x

### PART II

1. **(13 points)**- Student
`A`eats lunch at noon in the cafeteria once a week (including weekends) on a random schedule. Student`B`also eats there at noon, once a week (including weekends), on a random schedule. On average, how many times will`A`see`B`in a week? - If
`A`eats lunch in the cafeteria 3 times a week instead, and`B`eats there twice a week, how many times a week will they meet, on average? - If
`A`eats lunch in the cafeteria`m`times a week, and`B`eats there`n`times a week, how many times a week will they meet, on average?

2. **(13 points)**- Consider the following game: We are given a pile of
`n`jelly beans, where`n`is a positive integer. Two players alternately must eat 1 or 2 jelly beans off the pile. The winner is the person who eats the last jelly bean.For which values of

`n`is there a winning strategy for the beginner? Describe the strategy. - Consider a game involving a pile of
`n`jelly beans as above, except that the players alternately must eat 1, 3, or 5 jelly beans.For which values of

`n`is there a winning strategy for the beginner? Describe the strategy. - Consider a game involving a pile of
`n`jelly beans as above, except that the players alternately must eat 1, 2 or 4 jelly beans.For which values of

`n`is there a winning strategy for the beginner? Describe the strategy.

3. **(13 points)**`1, 2, 3, . . . , 20.`The die is ``loaded'', so that it is twice as likely to roll any given odd number as any given even number.- Find the probability of rolling a number greater than 10.
- Find the probability of rolling a number greater than 11.

4. **(13 points)**Inside square

`ABCD`(see figure) with sides of length 20 inches, segment`AE`is drawn, where`E`is the point on`DC`which is 6 inches from`D`. The perpendicular bisector of`AE`is drawn and intersects`AE, AD`, and`BC`at points`M, P`, and`Q`respectively. Find the ratio of segment`PM`to segment`MQ`. - Student