March 27, 1999

1. Do not open this exam until you are told to do so.

2. Print your name clearly on your answer sheet. On the "Grade" line, write

your grade in school (9,10,11, or 12)

3. Print clearly on the answer sheet your answer to each question. You should

not write decimal approximations to numbers such as ,and

You should simplify your answers as much as seems reasonable. For example

denominator. Frequently, several equivalent expressions will be considered

correct. For example,

4. Each correct answer is worth

worth

receive half of one point.

5. No books, notes, calculators or headphones may be used.

6. Most students will probably be unable to finish the exam in the 2 hours alloted.

fir the most party, the hardest questions are near the end of the exam.

7.

of a polygon, such as

around the polygon in a clockwise manner.

8. You may keep your questions and scratch work. You might want to keep a record

of your responses for comparison with the solutions, which will be distributed at 1:10

in Packard Lab rooms 416 and 466.

9. The people with the five highest scores of all people taking the test, without regard for

grade level, will receive cash awards of

In case of ties, the cash awards will be split. For example if two tie for first, they each

receive

10. A plaque will be awarded to the top individual and team in each grade level (9-12 for

individuals, 10-12 for teams). In case of a tie, duplicate awards will be presented.

2. If a triangle *ABC* has *AB = 3, AC = 4,* and *BC = 5,*
then what is the length of the altitude drawn from

vertex *A *to side* BC*?

3. If *f(x) = 2x ^{3}*, then

4. You have some quarters and dimes worth a total of *$7.15*. You
have three times as many dimes as

quarters. How many coins do you have?

5. The point *E* is outside the square *ABCD* so that *ABE*
is an equilateral triangle. What is the number

of degrees in the angle CED?

6. If *x, 3x + 1*, and *6x + 2* are in geometric progression,
then what is *x*?

10. You start with 128 cents, and six times bet half of your current
amount of money, with

probability .5 of winning on each occasion. You
win 3 times and lose 3 times in some order.

Does the amount of money which you have at the end
depend upon the order of wins and

losses, and if not, how many cents will you have
at the end? Your answer should be either the

word "depends" or else the number of cents that
you will always have at the end.

12. The number of pairs of positive integers *(x,y)* which satisfy
*2x
+ 3y = 515* is ?

13. Joe runs around the track at a pace of *80* seconds per lap.
Jim runs it in the opposite direction.

They meet every *30* seconds. How many seconds
does it take Jim to run each lap around the track?

14. Suppose that the angle between the minute hand and hour hand of
a clock is *60* degrees. If the minute

hand is *12* inches long and the hour
hand is *9* inches long, then what is the distance in inches between
the

tip ends of the hands?

15. If *x *and *y* are positive integers less than *100*
such that *x ^{2} - y^{2 }= 343*, then

16. Eight circular coins are arranged so that no two of them overlap
or touch, and no three

of them have a tangent line in common. What is the
total number of lines that are tangent to

two of the coins?

17. *AB* is a diameter of a circle of radius *1*. If *C*
is any point on the circle, then the possible

values of *AC + BC* are exactly those numbers
s which satisfy

18. When *17!* is written base 12, how many *0*s are at the
end?

19. An equaliateral triangle is incribed in a circle of radius *1*
with one vertex at the bottom of

the circle. What is the area that lies inside the circle and above
the triangle?

23. What positive integer *n* satisfies *log _{10}(225!)
- log_{10}(223!) = l + log_{10}(n!) *?

24. The sum of the first *40* terms of an arithmetic series equals
*300*,
and the sum of the next

*40 *terms equals *3500*. What is the
first term of the series?

25. For what real number *c *does the equation |*x ^{2}
+ 8x + 12*|

27. How many integers between *1000* and *9999* are perfect
squares and have sum of

digits equal to *14*?

28. For how many paths consisting of a sequence of a horizontal (forward
or backward)

and / or vertical line segments, each connecting
a pair of adjacent letter in the diagram below,

is the word "COUNT" spelled out as a path is traversed
from beginning to end?

29. Mary takes the train home from work each day, arriving at the station
at 6:00, at which time

her butler, Jeeves, meets her and drives her home.
Jeeves always drives at the same constant

speed, and always arrives at the station at exactly
6:00*.* One day Mary takes an earlier train

which arrives at the station at 5:00. She immediately
begins walking home. On the way, she

meets Jeeves, who is on his way to meet the* *6:00train.
Mary gets into the car, and they drive

home, arriving *20* minues earlier than usual.
How many minutes had Mary been walking?

30. Begin with a circle *C* with two smaller circles inscribed
in it. **Step 1:** Inscribe a circle into

each non circlular region of the resulting figure.
Now repeat Step 1 four more times. How

many of the resulting circles do not touch circle
*C
*?
The intital iteration and the first step are

shown below.

31. *25* people are arranged in a circle. Three are selected at
random. What is the

probablity that none of those selected were next
to one another?

32. How many pairs *(m,n)* of positive integers have the property
that the ratio of the

size of each interior angle of a regular *m*-gon
to that of a regular *n*-gon equals *3:2*?

34. What is the remainder when *6 ^{83} + 8^{83}*
is divided by

35. Let *C* be the circle of radius *1* centered at *(0,0)*,
Give, in simplifed form, a necessary and

sufficient condition on positive numbers *x _{1}*
and

36. The *36 ^{th}* digit after the decimal in the decimal
expansion of

37. Three people are playing a game. They take turns in rotation rolling
a fair *6*-sided die.

A player who rolls a *5* or *6* wins and the game ends. A
player who rollas a *1 *or* 2* loses and is

out of the game. The game continues until either someone wins (by rolling
a *5* or *6*) or else there is

only one person left, in which that person wins. What is the probability
that the first person to roll

wins?

38. Let *0,A, *and *B* denote the points *(0,0,0), (3,0,0)
and (0,4,0),* respectively. A sphere of

radius *2* in *xyz*-space is tangent to the lines *OA,
OB,* and *AB,* and the *z*-coordinate of its center

is positibe. That are the coordinates of the center of the sphere?

39. How many 4-digit numbers equal (a+b)^2, where a (resp. b) is the number formed by the first (resp. last) two digits of the 4-digit number.

40. Three circles are externally tangent to one another, and all lie
inside a circle of radius *4*, to

which they are all tangent. Two of the circles have radius *1 *and
*3*.
The center of these circles

lie along a diameter of the larger circle. What is the radius of the
third circle?