Question 1

Which of the following is the same as

Solution

Question 2

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pink pill, and Al's pills cost a total of for the two weeks. How much does one green pill cost?

Solution

Question 3

The sum of consecutive even integers is less than the sum of the first consecutive odd counting numbers. What is the smallest of the even integers?

Solution

Question 4

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost each, begonias each, cannas each, dahlias each, and Easter lilies each. What is the least possible cost, in dollars, for her garden?

Solution

Question 5

Moe uses a mower to cut his rectangular -foot by -foot lawn. The swath he cuts is inches wide, but he overlaps each cut by inches to make sure that no grass is missed. He walks at the rate of feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?

Solution

Question 6

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is . The horizontal length of a "-inch" television screen is closest, in inches, to which of the following?

Solution

Question 7

The symbolism denotes the largest integer not exceeding . For example, and . Compute

Solution

Question 8

The second and fourth terms of a geometric sequence are and . Which of the following is a possible first term?

Solution

Question 9

Find the value of that satisfies the equation

Solution

Question 10

Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of the three letters followed by three digits. By how many times is the number of possible license plates increased?

Solution

Question 11

A line with slope intersects a line with slope at point . What is the distance between the -intercepts of these two lines?

Solution

Question 12

Al, Betty, and Clare split among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of . Betty and Clare have both doubled their money, whereas Al has managed to lose . What was Al's original portion?

Solution

Question 13

Let denote the sum of the digits of the positive integer . For example, and . For how many two-digit values of is ?

Solution

Question 14

Given that where both and are positive integers, find the smallest possible value for .

Solution

Question 15

There are players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest players are given a bye, and the remaining players are paired off to play. After each round, the remaining players play in the next round. The tournament continues until only one player remains unbeaten. The total number of matches played is

Solution

Question 16

A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year ?

Solution

Question 17

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius? (Note: a cone with radius and height has volume and a sphere with radius has volume .)

Solution

Question 18

What is the largest integer that is a divisor of for all positive even integers ?

Solution

Question 19

Three semicircles of radius are constructed on diameter of a semicircle of radius . The centers of the small semicircles divide into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

Solution

Question 20

In rectangle and . Points and are on so that and . Lines and intersect at . Find the area of .

Solution

Question 21

A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?

Solution

Question 22

A clock chimes once at minutes past each hour and chimes on the hour according to the hour. For example, at there is one chime and at noon and midnight there are twelve chimes. Starting at on on what date will the chime occur?

Solution

Question 23

A regular octagon has an area of one square unit. What is the area of the rectangle ?

Solution

Question 24

The first four terms in an arithmetic sequence are and in that order. What is the ???fth term?

Solution

Question 25

How many distinct four-digit numbers are divisible by and have as their last two digits?

Solution

Answer Keys