Understanding AIME question formats

Below are some practice questions similar to what one could expect to find on the American Invitational Mathematics Examination (AIME). The AIME is a biannual, high-level invitational open to those with high scores on the American Mathematics Competition (AMC) 10 and/or AMC 12 tests. AIME exams cover a wide range of mathematical topics, including probability and algebra at the high school level, and consist of 15 questions to be answered over a 3-hour period.

Work your way through the eight problems below and, for an extra challenge, time yourself to see how fast you can solve these questions correctly. An important hint to remember about the AIME test is that each answer must be an integer between 0 and 999. Your responses should always be whole numbers, not fractions or irrational numbers. When you're ready to look, an answer key is provided at the bottom.

Problem 1

Nine students in the drama club are rehearsing an improv skit. Each member is assigned one of three roles: detective, criminal, or cop. At least one student is assigned to each role. Additionally, the number of students who were detectives was greater than the number who were criminals, which was greater than the number who were cops.

Let x be the number of different ways to assign the roles to the nine students. What is the remainder when x is divided by 1000?

Problem 2

There are three teams competing in a rowing competition. In a race covering a distance of x meters, Team A can beat Team B by 24 meters. Team B can beat Team C by 12 meters. If Team A can beat Team C by 33 meters, what is the length of the race, in meters?

Problem 3

Let a, b, and c be real numbers satisfying the system of equations:

ab + 3c = 42

bc + 3a = 42

ca + 3b = 42

Let T be the set of possible values of a. What is the sum of the elements in T?

Problem 4

At a summer art camp, 40% of the campers paint, 30% draw, 20% do digital art, and 10% sculpt. All painters are required to attend art theory class, and 80% of the drawers, 50% of the digital artists, and 20% of the sculptors choose to attend the theory class. If a camper is selected at random from those who attend theory class, the probability that they draw is ​m/n, where m and n are positive integers. What is the sum of m + n?

Problem 5

At a cooking club, 35% of the members bake, 25% grill, 25% do food prep, and 15% make desserts. All of the bakers are required to attend the weekly culinary seminar, while 60% of the grillers, 40% of the food preppers, and 20% of the dessert makers choose to attend. If a club member is randomly selected from those who attend the seminar, the probability that they are a griller can be expressed as m/n, where m and n are positive integers. What is the sum of m + n?

Problem 6

There are four different types of participants in an after-school theater program. 45% of members are actors, 25% are directors, 20% are writers, and 10% work on the stage crew. All actors are required to attend the weekly meeting with the theater owner. Additionally, 70% of the directors, 40% of the scriptwriters, and 10% of the lighting crew also attend this meeting. If a participant is randomly selected from those who attend the meeting, the probability that they are a director can be expressed as a reduced fraction m/n. Compute m + n.

Problem 7

Imagine that you are at a carnival. When wandering around the circus tents, you see a game that involves two identical spinning wheels. Each wheel is labeled with the numbers 1 through 6, but the spinner is weighted so that landing on number k is directly proportional to k. That means higher numbers are more likely to be landed on than lower ones. When both wheels are spun, what is the probability that the sum of the numbers they land on is 7?

Problem 8

How many different ordered pairs of positive integers (x, y) satisfy the following equation: 15 * x + 10y = 1450?

Closing: Important note on AIME answers

When submitting your official answers on the AIME test, remember that each response must be formatted as a three-digit integer. For AIME math problems with answers under 100, simply add a leading zero (for example, 045). Your questions will not be accepted if you do not follow the format required, so don’t let a small oversight cost you valuable points.

To score high on the AIME, you must dedicate significant time to mastering challenging AIME problems and practicing with mock questions and exams. For the questions above, ensure you understand how to arrive at the answers. Simply memorizing responses without grasping the underlying concepts will not adequately prepare you for test day. The effort is worth it: achievement on the AIME allows you to participate in the United States of America Math Olympiad (USAMO), the final qualification exam for the U.S. International Math Olympiad team. Stay focused, trust your preparation, and finish every math problem with confidence for a chance to compete with the best of the best in mathematics.

Answers

1. 16
2. 96
3. 11
4. 25
5. 26
6. 178
7. 71
8. 48