American Invitational Mathematics Exam (AIME)

By Eric Eng

By Eric Eng

a student pointing at the math problem on the board

How to Perform Well on the American Invitational Math Exam (AIME)

How do you perform well on the American Invitational Mathematics Exam (AIME) to qualify for the USA Math Olympiad (USAMO)? Performing well enough to qualify for the USAMO is an excellent award to show the depth of your intellectual capacity to the college admissions officers.

Learning isn’t just strictly hard work and raw talent. A lot of it is a mental game, which plays a huge role in the learning process. How good of a test taker are you? How confident are you in approaching problems you’re weak at or find difficult? Are you afraid of failure?

Very few students understand the fact that struggling on problems for long periods of time is actually how you learn. Only by struggling on a problem for extended periods of time do you challenge the neurons in your brain to fire in places they haven’t fired before, and make connections that facilitate in remarkable insights that you never previously discovered.

So don’t be afraid of failure. Take intellectual risks. Challenge yourself.

Struggling is a good thing. And if you can stomach your ego and accept that learning to fail is actually the best way to learn, then the sky is the limit. Because that’s what really separates the mediocre from the great learners.

a person taking a math test

To do well on the AIME, you need to first:

  1. Have a fundamental understanding of the subject matter
  2. Apply the fundamental understanding of the subject matter by practicing lots of problems

Most high students students who have only been exposed to their high school curriculum will not have a strong fundamental of the subject matter. This includes a strong grounding in discrete math – in particular combinatorics and number theory. Gain some strong number sense by understanding how to count things, understand the prime factors of numbers and the different ways in which they could be applied, 1–1 correspondences, and other counting techniques like the Balls in Urns formula. A strong combinatorial reasoning is crucial to perform well on the AIME.

Even the tough algebra and geometry problems will eventually require you to determine a discrete, integer solution using basic counting and number theory techniques. I feel that the mediocre students who have only been exposed to algebra and geometry in their school curriculum lack the combinatorics and number theory background to perform well enough on the AIME. If you know how to count and have strong number sense, you are in prime shape to tackle at least the first 10 problems on the AIME.

As for geometry, understand everything about the triangle, especially its relationship to the circle – the different ways to compute the area whether using Heron’s formula or the inradius and semiperimeter, including medians, angle bisectors, perpendicular bisectors, etc. Constantly search for similar triangles as that will solve 90% of the AIME geometry problems. Geometry requires practice and looking for clues. Draw a clear picture of the problem at hand, label angles and search for similar triangles that may lead you to your answer.

Student doing algebra on a board.

You can visit the Art of Problem Solving website and buy its V1 and V2 books to fill any gaps necessary to perform well on the American Invitational Mathematics Exam (AIME). There are numerous high performing amateur mathematicians who have written lengthy blogposts on how to do well. I would say V2 covers more topics can you possibly need – you only need to understand maybe half of the chapters in V2 really well in order to perform well enough on the AIME to advance to the USAMO.

Assuming you have the fundamental understanding of the subject matter, you need to do practice problems. You’ll be surprised that many of the AIME problems can be solved with a fundamental understanding of algebra, geometry, combinatorics and number theory – very rarely will you have to apply advanced theorems like Fermat’s Little Theorem to solve difficult problems.

The best problem solvers have a limited number of fundamental tools in their arsenal, but understand how to apply those fundamental tools really well. Not the other way around. Don’t fall into the trap of memorizing a whole slew of tools only to realize you still can’t solve problems. Assuming you have the basic tools in place, start solving and tackling hard problems and you will quickly see yourself improve.


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