Question 1
A basketball player made 
baskets during a game. Each basket was worth either 
or 
points. How many different numbers could represent the total points scored by the player?
Solution
Question 2
A 
block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
Solution
Question 3
A semipro baseball league has teams with 
players each. League rules state that a player must be paid at least 
dollars, and that the total of all players' salaries for each team cannot exceed 
dollars. What is the maximum possible salary, in dollars, for a single player?
Solution
Question 4
On circle 
, points 
and 
are on the same side of diameter 
, 
, and 
. What is the ratio of the area of the smaller sector 
to the area of the circle?
Solution
Question 5
)
A class collects 
dollars to buy flowers for a classmate who is in the hospital. Roses cost dollars each, and carnations cost dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly dollars?
Solution
Question 6
)
Postman Pete has a pedometer to count his steps. The pedometer records up to 
steps, then flips over to 
on the next step. Pete plans to determine his mileage for a year. On January 
Pete sets the pedometer to . During the year, the pedometer flips from to forty-four times. On December 
the pedometer reads 
. Pete takes 
steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
Solution
Question 7
)
For real numbers 
and 
, define $
. What is 
$
?
Solution
Question 8
)
Points 
and lie on 
. The length of is 
times the length of 
, and the length of 
is 
times the length of 
. The length of 
is what fraction of the length of ?
Solution
Question 9
)
Points 
and are on a circle of radius and 
. Point is the midpoint of the minor arc 
. What is the length of the line segment 
?
Solution
Question 10
)
Bricklayer Brenda would take hours to build a chimney alone, and bricklayer Brandon would take 
hours to build it alone. When they work together they talk a lot, and their combined output is decreased by bricks per hour. Working together, they build the chimney in hours. How many bricks are in the chimney?
Solution
Question 11
)
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top 
of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
Solution
Question 12
For each positive integer 
, the mean of the first terms of a sequence is . What is the 
th term of the sequence?
Solution
Question 13
Vertex 
of equilateral triangle 
is in the interior of unit square 
. Let 
be the region consisting of all points inside and outside whose distance from is between 
and 
. What is the area of ?
Solution
Question 14
)
A circle has a radius of 
and a circumference of 
. What is 
?
Solution
Question 15
)
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let be the region formed by the union of the square and all the triangles, and 
be the smallest convex polygon that contains . What is the area of the region that is inside but outside ?
Solution
Question 16
)
A rectangular floor measures by feet, where and are positive integers with 
. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair 
?
Solution
Question 17
)
Let , and be three distinct points on the graph of 
such that line is parallel to the 
-axis and 
is a right triangle with area . What is the sum of the digits of the 
-coordinate of ?
Solution
Question 18
)
A pyramid has a square base and vertex . The area of square is 
, and the areas of and 
are 
and 
, respectively. What is the volume of the pyramid?
Solution
Question 19
)
A function 
is defined by 
for all complex numbers 
, where 
and 
are complex numbers and 
. Suppose that 
and 
are both real. What is the smallest possible value of 
?
Solution
Question 20
)
Michael walks at the rate of feet per second on a long straight path. Trash pails are located every 
feet along the path. A garbage truck travels at feet per second in the same direction as Michael and stops for 
seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
Solution
Question 21
)
Two circles of radius 1 are to be constructed as follows. The center of circle is chosen uniformly and at random from the line segment joining 
and 
. The center of circle is chosen uniformly and at random, and independently of the first choice, from the line segment joining 
to 
. What is the probability that circles and intersect?
Solution
Question 22
)
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
Solution
Question 23
)
The sum of the base- logarithms of the divisors of 
is 
. What is ?
Solution
Question 24
)
Let 
. Distinct points 
lie on the -axis, and distinct points 
lie on the graph of 
. For every positive integer , 
is an equilateral triangle. What is the least for which the length 
?
Solution
Question 25
)
Let be a trapezoid with 
, 
, 
, 
, and 
. Bisectors of 
and 
meet at 
, and bisectors of 
and 
meet at 
. What is the area of hexagon 
?
Solution
Answer Keys
Question 1: E
Question 2: B
Question 3: C
Question 4: D
Question 5: C
Question 6: A
Question 7: A
Question 8: C
Question 9: A
Question 10: B
Question 11: A
Question 12: B
Question 13: B
Question 14: C
Question 15: C
Question 16: B
Question 17: C
Question 18: E
Question 19: B
Question 20: B
Question 21: E
Question 22: E
Question 23: A
Question 24: C
Question 25: B