Question 1

What is

Solution

Question 2

At the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost?

Solution

Question 3

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

Solution

Question 4

Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?

Solution

Question 5

On an algebra quiz, of the students scored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?

Solution

Question 6

The difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse?

Solution

Question 7

The first three terms of a geometric progression are , , and . What is the fourth term?

Solution

Question 8

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: off the listed price if the listed price is at least

Coupon 2: dollars off the listed price if the listed price is at least

Coupon 3: off the amount by which the listed price exceeds

For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?

Solution

Question 9

Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?

Solution

Question 10

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

Solution

Question 11

David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?

Solution

Question 12

Two circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?

Solution 1

This is the diagram for you to see the solution.

Solution 2

Question 13

A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than friends per room. In how many ways can the innkeeper assign the guests to the rooms?

Solution

Question 14

Let be three integers such that is an arithmetic progression and is a geometric progression. What is the smallest possible value of ?

Solution

Question 15

A five-digit palindrome is a positive integer with respective digits , where is non-zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of ?

Solution

Question 16

The product , where the second factor has digits, is an integer whose digits have a sum of . What is ?

Solution

Question 17

A rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?

Solution

Question 18

The domain of the function is an interval of length , where and are relatively prime positive integers. What is ?

Solution

Question 19

There are exactly distinct rational numbers such that and has at least one integer solution for . What is ?

Solution

Question 20

In , , , and . Points and lie on and respectively. What is the minimum possible value of ?

Solution

Question 21

For every real number , let denote the greatest integer not exceeding , and let The set of all numbers such that and is a union of disjoint intervals. What is the sum of the lengths of those intervals?

Solution

Question 22

The number is between and . How many pairs of integers are there such that and

Solution

Question 23

The fraction where is the length of the period of the repeating decimal expansion. What is the sum ?

Solution

Question 24

Let , and for , let . For how many values of is ?

Solution

Question 25

The parabola has focus and goes through the points and . For how many points with integer coordinates is it true that ?

Solution

Answer Keys