Question 1
A large urn contains 
balls, of which 
are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be 
? (No red balls are to be removed.)
Solution
Question 2
While exploring a cave, Carl comes across a collection of 
-pound rocks worth 
each, 
-pound rocks worth 
each, and 
-pound rocks worth 
each. There are at least 
of each size. He can carry at most 
pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
Solution
Question 3
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
Solution
Question 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let 
be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
Solution
Question 5
What is the sum of all possible values of 
for which the polynomials 
and 
have a root in common?
Solution
Question 6
For positive integers 
and 
such that 
, both the mean and the median of the set 
are equal to . What is 
?
Solution
Question 7
For how many (not necessarily positive) integer values of is the value of 
an integer?
Solution
Question 8
All of the triangles in the diagram below are similar to isosceles triangle 
, in which 
. Each of the 7 smallest triangles has area 1, and 
has area 40. What is the area of trapezoid 
?
Solution
Question 9
Which of the following describes the largest subset of values of 
within the closed interval 
for which 
for every 
between 
and 
, inclusive? 
Solution
Question 10
How many ordered pairs of real numbers 
satisfy the following system of equations? 
Solution
Question 11
A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point 
falls on point 
. What is the length in inches of the crease? 
Solution
Question 12
Let 
be a set of 6 integers taken from 
with the property that if 
and 
are elements of with 
, then is not a multiple of . What is the least possible value of an element in 
Solution
Question 13
How many nonnegative integers can be written in the form 
where 
for 
?
Solution
Question 14
The solutions to the equation 
, where is a positive real number other than 
or 
, can be written as 
where 
and 
are relatively prime positive integers. What is 
?
Solution
Question 15
A scanning code consists of a 
grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 
squares. A scanning code is called 
if its look does not change when the entire square is rotated by a multiple of 
counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Solution
Question 16
Which of the following describes the set of values of for which the curves 
and 
in the real 
-plane intersect at exactly 
points?
Solution
Question 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?
Solution
Question 18
Triangle with 
and 
has area 
. Let 
be the midpoint of 
, and let 
be the midpoint of 
. The angle bisector of 
intersects 
and 
at 
and 
, respectively. What is the area of quadrilateral 
?
Solution
Question 19
Let be the set of positive integers that have no prime factors other than 
, , or . The infinite sum 
of the reciprocals of the elements of can be expressed as 
, where and are relatively prime positive integers. What is ?
Solution
Question 20
Triangle is an isosceles right triangle with 
. Let 
be the midpoint of hypotenuse . Points 
and lie on sides and , respectively, so that 
and 
is a cyclic quadrilateral. Given that triangle 
has area , the length 
can be written as 
, where , , and 
are positive integers and is not divisible by the square of any prime. What is the value of 
?
Solution
Question 21
Which of the following polynomials has the greatest real root? 
Solution
Question 22
The solutions to the equations 
and 
where 
form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form 
where 
and 
are positive integers and neither nor is divisible by the square of any prime number. What is 
Solution
Question 23
In 
and 
Points 
and lie on sides 
and 
respectively, so that 
Let and 
be the midpoints of segments 
and 
respectively. What is the degree measure of the acute angle formed by lines 
and 
Solution
Question 24
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between and 
Armed with this information, what number should Carol choose to maximize her chance of winning?
Solution
Question 25
For a positive integer and nonzero digits , , and , let 
be the -digit integer each of whose digits is equal to ; let 
be the -digit integer each of whose digits is equal to , and let 
be the 
-digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of 
for which there are at least two values of such that 
?
Solution
Answer Keys
Question 1: D
Question 2: C
Question 3: E
Question 4: D
Question 5: E
Question 6: B
Question 7: E
Question 8: E
Question 9: E
Question 10: C
Question 11: D
Question 12: C
Question 13: D
Question 14: D
Question 15: B
Question 16: E
Question 17: D
Question 18: D
Question 19: C
Question 20: D
Question 21: B
Question 22: A
Question 23: E
Question 24: B
Question 25: D