1. By how much does 1/3 of 5/2 exceed 1/2 of 1/3 ?
2. What fraction of the area of a circle of radius 5 lies between radius 3
and radius 4?
3. A ticket fee was $10, but then it was reduced. The number of customers
increased by
50%, but the amount of money received only increased by 20%. How many dollars
was the reduced ticket price?
4. If 6x + 7y = 2007 and 7x + 6y = 7002, then what is the value of x + y?
5. What is the sum of the digits of 1055 - 55?
6. There are three consecutive positive integers such that the square of the
second minus
twelve times the first is three less than twice the third. What is the smallest
of the
three integers?
7. How many perfect cubes greater than 1 are divisors of 99?
8. The harmonic mean of two numbers is defined to be the reciprocal of the
average of
the reciprocals of the numbers. Find x such that the harmonic mean of 4 and x is
6.
9. What is the smallest possible value of
for any real numbers
x and y?
10. What is the value of log2(log2(log2(16)))?
11. Two fair 6-sided dice, colored red and white, are tossed. What is the
probability that
the number on the red die is at least as large as that on the white die?
12. What is the smallest positive number which is equal to the cube of one
positive integer
and also is equal to the fourth power of a different positive integer?
13. What value of c occurs in a solution of the system of equations a+b+c =
14, ab = 14,
and c2 = a2 + b2?
14. If 11 positive integers
have the property that no three of them
are the sides of a triangle, what is the smallest possible value of
15. A cube and sphere have the same surface area. What is the ratio of the
volume of the
sphere to that of the cube?
16. A rhombus ABCD with sides of length 5 lies in the first quadrant with A
at the
origin. If AD has slope 1/2 and AB has slope 2, then what are the coordinates of
C?
17. For how many primes p is p2 + 3p - 1 also prime?
18. A large container, labeled R, is partially filled with 4 quarts of red
paint. Another
large container, labeled W, is partially filled with 5 quarts of white paint. A
small
empty can is completely filled with red paint taken from R, and the contents of
the
can then emptied into W. After thorough mixing of the contents of W, the can is
completely filled with some of this mixture from W, and the contents of the can
then
emptied into R. The ratio of red paint to white in R is now 3.1. What is the
size of
the can, in quarts?
19. What is the largest number A such that the graphs of x2 = y2
and (x - A)2 + y2 = 1
intersect?
20. A number is written with one 1, followed by three 3’s, then five 5’s,
then seven 7’s,
then nine 9’s, then eleven 11’s, etc. Thus it begins 13335555577. .
. , and just
as it gets
to the 13’s it has . . . 11131313. . . . What is its 999th digit?
21. What is the radius of a circle inscribed in an isosceles right triangle
whose legs have
length 1?
22. In what base b is the equation 53*15 = 732 valid? Here all three numbers
are base-b
numbers, and b must be a positive integer.
23. A strip of rubber is initially 80 cm long, and after every minute it is
instantaneously
and uniformly stretched by 40 cm. An ant moves at a rate of 42 cm per minute,
starting at one end. Each time the strip is stretched, the ant is moved to a
position
on the modified strip proportional to its position on the strip before the
stretching
took place. How many minutes does it take the ant to cross the strip?
24. The first term of an infinite geometric series is 10, and the sum of the
series lies
between 9 and 11, inclusive. What is the range of values for the common ratio r
between terms of this series?
25. If the roots of x 2 - bx + c = 0 are sin(π/9)
and cos(π/9), then express b in terms of
c. (Don’t write “b =”; just write the expression involving c.)
26. Let R denote the set of points (x, y) satisfying
What is the
area of R?
27. The sum of the 3-digit numbers 35x and 4y7 is divisible by 36. Find all
possible
ordered pairs (x, y).
28. What is the ratio of the area of a regular 10-gon to that of a regular
20-gon inscribed
in the same circle? Express your answer using a single trig function with its
angle in
degrees.
29. Let k > 0 and let A lie on the curve
so that the vertical and horizontal
segments from A to the x- and y-axes are sides of a square with the origin at
the
vertex opposite A. Let B be the vertex of this square lying below A on the
x-axis.
Let C be the point on the same curve between the origin and A such that the
vertical
segment from C to the x-axis and the horizontal segment from C to AB is a square
with B at the vertex opposite C. What is the ratio of the side length of the
second
square to that of the first?
30. What is the smallest multiple of 999 which does not have any 9’s among
its digits?
(If the number is n = 999 * d, write n, not d.)
31. Let P(n) denote the product of the digits of n, and let S(n) denote the
sum of the
digits of n. How many positive integers n satisfy n = P(n) + S(n)?
32. For each point in the plane, consider the sum of the squares of its
distances from the
four points (-3,-1), (-1, 0), (1, 2), and (1, 3). What is the smallest number
achieved
as such a sum?
33. What is the remainder when
is divided by x2 - 1?
34. Find k so that the solutions of x3 - 3x2 + kx + 8 = 0 are in arithmetic
progression .
35. In triangle ABC, D lies on BC so that AC = 3, AD = 3, BD = 8, and CD = 1.
What is the length of AB?
36. What is the ninth digit from the right in 10120?
37. Two circles of radius 1 overlap so that the center of
each lies on the circumference of
the other. What is the area of their union?
38. The first two positive integers n for which 1+2+...+n
is a perfect square are 1 and
8. What are the next two?
39. In the diagram below, which is not drawn to scale, the
circles are tangent at A, the
center of the larger circle is at B, CD = 42, and EF = 24. What are the radii of
the
circles?
40. What is the side length of an equilateral triangle ABC
for which there is a point P
inside it such that AP = 6, BP = 8, and CP = 10?
