Question 1
What is 
Solution
Question 2
Josanna's test scores to date are 
and 
. Her goal is to raise her test average at least 
points with her next test. What is the minimum test score she would need to accomplish this goal?
Solution
Question 3
At a store, when a length is reported as 
inches that means the length is at least 
inches and at most 
inches. Suppose the dimensions of a rectangular tile are reported as 
inches by inches. In square inches, what is the minimum area for the rectangle?
Solution
Question 4
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid 
dollars and Bernardo had paid 
dollars, where 
. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
Solution
Question 5
In multiplying two positive integers 
and 
, Ron reversed the digits of the two-digit number . His erroneous product was 
. What is the correct value of the product of and ?
Solution
Question 6
On Halloween Casper ate 
of his candies and then gave candies to his brother. The next day he ate of his remaining candies and then gave 
candies to his sister. On the third day he ate his final 
candies. How many candies did Casper have at the beginning?
Solution
Question 7
The sum of two angles of a triangle is 
of a right angle, and one of these two angles is 
larger than the other. What is the degree measure of the largest angle in the triangle?
Solution
Question 8
At a certain beach if it is at least 
and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
Solution
Question 9
The area of 
is one third of the area of 
. Segment 
is perpendicular to segment 
. What is 
?
Solution
Question 10
Consider the set of numbers 
. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
Solution
Question 11
There are 
people in a room. What is the largest value of 
such that the statement "At least people in this room have birthdays falling in the same month" is always true?
Solution
Question 12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of 
meters, and it takes her 
seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
Solution
Question 13
Two real numbers are selected independently at random from the interval 
. What is the probability that the product of those numbers is greater than zero?
Solution
Question 14
A rectangular parking lot has a diagonal of 
meters and an area of 
square meters. In meters, what is the perimeter of the parking lot?
Solution
Question 15
Let 
denote the "averaged with" operation: 
. Which of the following distributive laws hold for all numbers 
and 
? 
Solution
Question 16
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
Solution
Question 17
In the given circle, the diameter 
is parallel to 
, and 
is parallel to 
. The angles 
and 
are in the ratio 
. What is the degree measure of angle 
?
Solution
Question 18
Rectangle 
has 
and 
. Point 
is chosen on side so that 
. What is the degree measure of 
?
Solution
Question 19
What is the product of all the roots of the equation 
Solution
Question 20
Rhombus has side length and 
??. Region 
consists of all points inside the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?
Solution
Question 21
Brian writes down four integers 
whose sum is 
. The pairwise positive differences of these numbers are 
and 
. What is the sum of the possible values for 
?
Solution
Question 22
A pyramid has a square base with sides of length 
and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Solution
Question 23
What is the hundreds digit of 
?
Solution
Question 24
A lattice point in an 
-coordinate system is any point 
where both and 
are integers. The graph of 
passes through no lattice point with 
for all 
such that 
. What is the maximum possible value of ?
Solution
Question 25
Let 
be a triangle with sides 
and 
. For 
, if 
and 
and 
are the points of tangency of the incircle of 
to the sides 
and 
respectively, then 
is a triangle with side lengths 
and 
if it exists. What is the perimeter of the last triangle in the sequence 
?
Solution
Answer Keys
Question 1: C
Question 2: E
Question 3: A
Question 4: C
Question 5: E
Question 6: A
Question 7: B
Question 8: B
Question 9: D
Question 10: B
Question 11: D
Question 12: A
Question 13: D
Question 14: C
Question 15: E
Question 16: A
Question 17: C
Question 18: E
Question 19: A
Question 20: C
Question 21: B
Question 22: A
Question 23: D
Question 24: B
Question 25: D