How do I qualify for the AIME?

The American Invitational Mathematics Exam (AIME) is an intermediate exam in the American Math Competition (AMC) sequence, after the AMC 10 or 12 but before the USAJMO and USAMO. Qualification for the AIME is a great achievement, and shows that you are an avid math student and capable problem solver.

Who qualifies for AIME?

The MAA invites at least the top 2.5% of AMC 10 scorers and at least the top 5% of AMC 12 scorers. However, in recent years, the MAA has invited many more students, making the AIME floor usually around the top 6% of the AMC 10 and the top 12% of the AMC 12. In recent years, the AIME cutoff for the AMC 10 has ranged from 94.5 to 105, and 85.5 to 93 for the AMC 12.

AMC 10 vs. AMC 12

If you are a younger student who's goal is to simply qualify for the AIME, you may want to take the AMC 12 for lower cutoffs. The AMC 10 and 12 overlap by 11-14 questions each year, meaning that the jump in difficulty between the AMC 10 and AMC 12 is quite manageable, especially for the first 15 problems. If you have not learned the prerequisite math for the AMC 12(Trigonometry and Precalculus), I recommend you stick to the AMC 10. Both the AMC 10 and AMC 12 follow a similar topic distribution, as shown below.

Where do I start?

If you have no idea where to start in this journey, that's okay! The book Volume One by Sandor Lehoczky and Richard Rusczyk is the perfect introduction to competition math, exploring introductory-level ideas in a manner that is both systematic and easy to follow. After finishing Volume One, I recommend the book Intro to Counting & Probability by David Patrick, which should give you the tools to approach nearly all AMC 10/12 Combinatorics problems, which make up 25% of the test! The concepts covered in these two books are more than sufficient for AIME qualification through both the AMC 10 and 12 if you fully understand each chapter.

Some tips from my personal experience:

• Algebra is overrated: While Algebra problems take up around 40% of the competition, you can probably solve most of them already. This is because Algebra and algebraic word problems are taught in school, which gives you the tools to approach simpler Algebra problems. On the other hand, Combinatorics, which consists of Counting and Probability, is rarely, if ever, taught in schools.

• Don't just read it, do it: Reading about math concepts is great, but without practice, you won't know how to apply them in a competition. For example, on Problem 13 of the 2024 AIME I, I could physically visualize Fermat's Little Theorem in my notebook, but I overlooked its application because I simply had not done enough problems for me to recognize it, leading me to skip the problem.

• Mock frequently and rigorously: Taking timed past tests in simulated contest conditions every other week will allow you to see your progress over time. It is important NOT to use your phone, use a calculator, use Desmos, etc as you will not have these during the test! Taking accurate mock contests is crucial as they enable you to identify shortcomings and weaknesses, which you can fix before the actual competition. You should compare your mock test score with the actual cutoff for the AIME that year, which can be found here.

• Use your answer choices: The AMC 10 and 12 have answer choices for a reason. If you are stumped about a problem, plugging in answer choices may not be a bad option. Eliminate obviously wrong answers. If you know the answer is a multiple of three and "2" is an answer choice, then it is obviously incorrect. Remember that this should not be your strategy of choice, and if it happens during a mock contest, make sure to learn the proper way to solve the problem after.

• Bring a ruler: The AMC 10/12 permits you to bring a ruler and compass. Drawing accurate, to-scale diagrams can allow you to see smart constructions, and you can literally measure to find the answer in a last resort.

• Don't guess unless you can eliminate at least two answer choices with certainty: The AMC 10/12 issues 1.5 points for blank answers, but zero for wrong ones. Your expected value is 1.2 points when guessing at random, and 1.5 if you can eliminate at least one choice. However, it is unlikely that you can eliminate an answer with 100% certainty, dropping your expected value to below 1.5. Eliminating two or more gives an expected value of 2 or more points per guess.

• Finally, have fun! You should enjoy the intellectual challenges that you encounter on this journey.

Good luck! Feel free to leave any questions in the comment section below.