Qualifying for the American Invitational Mathematics Examination is a milestone, and the next question every student asks is simple: what exactly do I need to study?
The AIME sits between the AMC and the national olympiads. It is an invitation-only exam administered by the Mathematical Association of America (MAA), and it rewards deep problem-solving rather than broad memorization. Understanding the AIME syllabus helps you spend prep time where it counts. Below, BIAA breaks down the format and the core topics so you can build a focused study plan.
How the AIME Works
The AIME is a 15-question, 3-hour exam. Every answer is an integer from 0 to 999, so there are no multiple-choice options and no answer to guess from a list. You earn one point per correct answer with no penalty for wrong answers, which means a final score is an integer from 0 to 15.
Students typically qualify by ranking near the top of the AMC 10 or AMC 12, with the AMC 10 path requiring a higher percentile. There is also a pathway through the USAMTS, a free proof-based contest. Two versions are offered each cycle, the AIME I and AIME II, with scheduling rules that differ for domestic and international students.
Qualification cutoffs, exact dates, fees, and the USAMO/USAJMO index formula change between cycles. Always confirm current details on the official MAA website before you register.
The AIME Syllabus: Core Topic Areas
The AIME draws from standard high-school mathematics, but it combines ideas in ways that demand real ingenuity. The syllabus centers on five interconnected areas.
Algebra
Algebra is the backbone of the exam. Expect polynomials (including the binomial theorem, Vieta's formulas, symmetric polynomials, conjugate roots, and polynomial division), sequences and series, functional relationships, and complex numbers. Many of the most elegant AIME problems hinge on recognizing an algebraic structure hidden inside a word problem.
Geometry
Geometry questions cover triangles, circles, quadrilaterals, and the relationships among them, often blended with trigonometry. Strong students are comfortable with the law of sines and cosines, power of a point, similar triangles, and coordinate methods. Because answers must be integers, problems are frequently engineered so that a clever setup collapses to a clean numeric result.
Number Theory
This area includes divisibility, prime factorization, and modular arithmetic. Number theory rewards students who can reason about remainders and structure rather than grind through arithmetic, and it appears often in the integer-answer format the AIME uses.
Combinatorics and Probability
Counting techniques, permutations and combinations, binomial coefficients, and probability round out the syllabus. These problems test careful case analysis and the ability to count without double-counting, a skill that overlaps heavily with discrete math used in competitive programming.
How Difficulty Is Structured
AIME problems generally escalate in difficulty from question 1 to question 15. The opening problems are roughly at the level of a hard AMC 12 question, while the final few are genuinely olympiad-adjacent and can stump even experienced students. A smart strategy is to secure every accessible problem first, then invest remaining time in the harder back half.
The AIME is less about how much you know and more about how flexibly you can apply a small set of powerful ideas under time pressure.
Building an Effective Study Plan
Because the topics interlock, scattered review is rarely effective. A structured approach works far better:
- Master the fundamentals first. Make sure your AMC-level algebra, geometry, and counting are automatic before chasing advanced techniques.
- Practice with real past papers. Timed sets of authentic AIME problems teach pacing and reveal recurring patterns far better than random worksheets.
- Review by topic, then by mistake. After each practice exam, sort missed problems by topic area to find your weakest category, then drill it deliberately.
- Connect topics. Notice how a number-theory idea solves a counting problem, or how complex numbers simplify a geometry question. These bridges are where AIME points live.
Consistent, deliberate practice with feedback is what separates a qualifier from a high scorer. Students who treat preparation as a long-term skill-building project, not a last-minute cram, are the ones who move on toward the USAMO and USAJMO.
If you want a guided path from AMC qualification through AIME mastery, explore BIAA's competition math program or browse our full range of academic competitions to find the right next challenge for your student.