1. Which of the following the following is the largest?
a. 234, b. 432, c. 842, d. 1682, e. 842
2. Rank the following collections of round pizzas in increasing order of
area:
(A) five 10-inch pizzas; (B) three 14-inch pizzas; (C) two 16-inch pizzas.
a. BAC, b. CAB, c. CBA, d. ABC, e. ACB
3. A computer virus is eating disk space. During the first day
it eats 1/2 of the disk space. During the second day it eats 1/3 of the
remaining disk space. The third day
it eats 1/4 of what still remains and the fourth day it eats 1/5
of what is left.
What fraction of the original disk space remains uneaten?
a. 1/5, b. 1/6, c. 1/10, d. 1/12, e. 1/24
4. Walter Wildcat runs 3 times as fast as Timothy Terrapin walks. Suppose
that Timothy, who is smarter than Walter, finishes an exam at 2:00 and
begins walking home. Walter finishes at 2:12 and runs after Timothy. At
what time does Walter catch up to Timothy?
a. 2:15, b. 2:16, c. 2:17, d. 2:18, e. 2:19
5. An integer p is prime if p>=2 and the only divisors of p are 1
and p.
Let M be the product of the first 100 primes.
How many 0's are at the end of M?
a. 0, b. 1, c. 10, d. 20, e. 100
6. Five sheep can eat the grass of 5 football fields
in 5 days. How many days would it take 3 sheep to eat the grass
of 3 football fields?
a. 2, b. 3, c. 4, d. 5, e. 6
7. Three little pigs open e-trade accounts and become day traders. After hard work they
achieve the following steady rates of losing money: the first one loses $1000 an hour,
the second one loses $1000 in two hours, and the third one loses $1000 in three hours.
The number of minutes it takes all three of them together to lose $2000 is between
a. 180 and 270, b. 120 and 180, c. 90 and 120, d. 66 and 90, e. 60 and 66
8. The grand prize in a basketball tournament is the largest collection of the new
golden dollar coins that can be placed on a 50 by 94 foot basketball court without
overlapping. The new dollar coin is 1.04 inches in diameter. The prize
is worth between
a. $10 and $10,000, b. $10,000 and $40,000, c. $40,000 and $200,000,
c. $200,000 and $1 million, d. $1 million and $10 million
9. In triangle ABC, angle C is three times as big as angle A and angle B is twice as big
as angle A.
The ratio of the length of AB to BC is
a. 1/3, b. 1/2, c. 2/3, d. 31/2, e. 2
10. Of the seven points (9,17), (6,11), (3,5), (7,12), (3.5,6), (5,10), (5,9), five lie on a straight line.
Which two do NOT lie on that line?
a. (5,10) and (7,12), b. (3,5) and (5,9), c. (9,17) and (7,12),
d. (6,11) and (3,5), e. (3.5,6) and (5,9)
11. Let a,b,c be nonzero integers.
Let p(x)=ax2 + bx+c.
Which of the following CANNOT occur?
a. p(x) has only one root and that root is positive
b. p(x) has only one root and that root is negative
c. p(x) has only one root and that root is rational
d. p(x) has only one root and that root is irrational
e. p(x) has no real roots
12. How many pairs of positive integers (a,b) are there
with a<=b and 1/6=1/a+1/b?
a. 1, b. 2, c. 3, d. 4, e. 5
13. Let A be the area of a circle that has radius (pi)1/2.
Within which interval does A lie?
a. 0< A<1, b. 1< A<4, c. 4< A<9, d. 9< A<16, e. 16< A<25
14. Ten people play a round-robin chess tournament (every person plays everyone else
exactly once). In each game, the winner gets 3 points, the loser gets 0 points, and in
case of a draw (tie), each of the two players gets 1 point.
The total number of points given to all players is 130.
How many games are draws?
a. 1, b. 2, c. 3, d. 4, e. 5
15. Berries are 90% water and 10% pulp, measured by weight, when freshly
picked. If the berries sit for a week, part of the water evaporates and the
berries become 80% water and 20% pulp.
If 20 pounds of freshly picked
berries sit for a week, how much will they weigh?
a. 8 pounds b. 10 pounds c. 12 pounds d. 15 pounds e. 18 pounds
16. The lower two vertices of a square lie on the x-axis and the upper two
vertices of the square lie
on the parabola y=15-x2. What is the area of the square?
a. 9, b. 10(2)1/2, c. 16, d. 25, e. 36
17. The sum of several (not necessarily different) positive integers is 10.
What is the largest possible value of their product?
a. 25, b. 32, c. 36, d. 54, e. 64
18. Order sin1, sin2, sin3 from smallest to largest (the angles are measured
in radians).
a. sin1‹sin2‹sin3, b. sin3‹sin2‹sin1,
c. sin1‹sin3‹sin2, d. sin2‹sin1‹sin3, e. sin3‹sin1‹sin2
19. The lines AB and CD are parallel and a distance 4 apart.
Suppose that AD intersects BC at a point P between
the lines. If AB=4 and CD=12, how far is P from the line CD?
a. 1, b. 2, c. 3, d. 4, e. 6
20. The triangular numbers are the numbers of the form n(n+1)/2,
where n=1, 2, 3, ...
How many of the first 100 triangular numbers end with a 0?
a. 10, b. 20, c. 30, d. 40, e. 50
21. Let a and b be real numbers with a>1 and b >1. Let loga(b)
denote the logarithm of b for the base a.
Four students compute
the product loga(b)·logb(a).
The first gets the answer 1. The second gets logab(ab)$. The third
gets abba. The fourth gets
logb(a)+loga(b).
How many of thestudents are correct?
a. 0, b. 1, c. 2, d. 3, e. 4
22. A student writes 44·55=3506. The student is calculating
in base b and has written a correct equation. What is b?
a. 7, b. 8, c. 9, d. 12, e. 14
23. Horace always lies. One day he told his neighbor Frank
``At least one of us never lies.'' From this information it is certain that
a. Frank always lies. b. Frank sometimes lies. c. Frank always tells the
truth. d. Frank sometimes tells the truth. e. Frank has never said anything.
24. Suppose f(x+1,y)=f(x,y)+y+1, f(x,0)=x, f(x,y)=f(y,x) for all x, y. What
is f(12,5)?
a. 5, b. 6, c. 12, d. 17, e. 77
25. At 1:00 pm on October 25, 2000, a dog falls asleep. It sleeps for 5 hours, then is
awake for 5 hours, then sleeps for 5 hours, etc.
At 1:17 pm on the same day, a cat falls asleep. It sleeps for 2 hours, then is awake
for 2 hours, then sleeps for 2 hours, etc.
Determine the largest integer H so that in every unbroken 24 hour period
in the year 2001 there
must be a total of at least H hours during which both animals will be asleep.
a. 4, b. 5, c. 6, d. 7, e. 10