Question 1

A bakery owner turns on his doughnut machine at . At the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

Solution

Question 2

What is the reciprocal of ?

Solution

Question 3

Suppose that of bananas are worth as much as oranges. How many oranges are worth as much as of bananas?

Solution

Question 4

Which of the following is equal to the product

Solution

Question 5

Suppose that is an integer. Which of the following statements must be true about ?

Solution

Question 6

Heather compares the price of a new computer at two different stores. Store offers off the sticker price followed by a rebate, and store offers off the same sticker price with no rebate. Heather saves by buying the computer at store instead of store . What is the sticker price of the computer, in dollars?

Solution

Question 7

While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?

Solution

Question 8

What is the volume of a cube whose surface area is twice that of a cube with volume 1?

Solution

Question 9

Older television screens have an aspect ratio of . That is, the ratio of the width to the height is . The aspect ratio of many movies is not , so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of and is shown on an older television screen with a -inch diagonal. What is the height, in inches, of each darkened strip?

Solution

Question 10

Doug can paint a room in hours. Dave can paint the same room in hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by ?

Solution

Question 11

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the visible numbers have the greatest possible sum. What is that sum?

Solution

Question 12

A function has domain and range . (The notation denotes .) What are the domain and range, respectively, of the function defined by ?

Solution

Question 13

Points and lie on a circle centered at , and . A second circle is internally tangent to the first and tangent to both and . What is the ratio of the area of the smaller circle to that of the larger circle?

Solution

Question 14

What is the area of the region defined by the inequality ?

Solution

Question 15

Let . What is the units digit of ?

Solution

Question 16

The numbers , , and are the first three terms of an arithmetic sequence, and the term of the sequence is . What is ?

Solution

Question 17

Let be a sequence determined by the rule if is even and if is odd. For how many positive integers is it true that is less than each of , , and ?

Solution

Question 18

A triangle with sides , , is placed in the three-dimensional plane with one vertex on the positive axis, one on the positive axis, and one on the positive axis. Let be the origin. What is the volume of ?

Solution

To solve this problem, first draw the 3-d coordinates.
Using Pythagorean Theorem, for triangles BOC, AOC, and AOB, you will be able to find out the values of a, b, and c, to be square root of 19, 6, and 30.
Now, notice BCOA is a right-triangle pyramid. What is the rule for calculating volume of a pyramid? It is 1/3 of height * base area.
What is the area of base AOC? It is 1/2 a*c; What is the height BO? It is b.

Question 19

In the expansion of what is the coefficient of ?

Solution

Question 20

Triangle has , , and . Point is on , and bisects the right angle. The inscribed circles of and have radii and , respectively. What is ?

Solution

Question 21

A permutation of is heavy-tailed if . What is the number of heavy-tailed permutations?

Solution

Question 22

A round table has radius . Six rectangular place mats are placed on the table. Each place mat has width and length as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is ?

Solution

Question 23

The solutions of the equation are the vertices of a convex polygon in the complex plane. What is the area of the polygon?

Solution

Question 24

Triangle has and . Point is the midpoint of . What is the largest possible value of ?

Solution

Question 25

A sequence , , , of points in the coordinate plane satisfies

for .

Suppose that . What is ?

Solution

Answer Keys