Mathematics Olympiad Preparation: A Practical Guide for 2025

The American Mathematics Competition sequence runs: AMC 8 → AMC 10/12 → AIME → USAMO/AMC Junior (USAJMO). Each level requires different skills and different preparation strategies.

AMC 8 preparation (target: top 5%, score ≥ 19/25)

The AMC 8 is a 25-question, 40-minute test with no penalty for wrong answers. The content is middle-school mathematics: arithmetic, fractions, geometry (area and perimeter), basic probability, and combinatorics.

The single most effective preparation strategy at this level is speed. The mathematics is accessible to most students who have completed Grade 7; the constraint is finishing 25 questions in 40 minutes while solving novel problems that require careful reading.

Recommended: AMC 8 past papers from 2015–2024 (available free at the Art of Problem Solving library). Complete each under timed conditions. Review every problem you could not solve within 2 minutes, not just those you answered incorrectly.

AMC 10/12 and AIME preparation

The AMC 10 and 12 require substantially different mathematics: number theory, algebra (functions, polynomials, sequences), geometry (including circles, trigonometry in AMC 12), and combinatorics at a higher level.

The most productive preparation resource at this level is the Art of Problem Solving textbook series: Introduction to Number Theory, Introduction to Counting and Probability, and Precalculus. These are not reference books — they require active problem-solving. Students who read them passively find that the knowledge does not transfer to competition conditions.

The AIME requires proof-like reasoning and multi-step constructions. Students who can score 10+ on AMC 10/12 but cannot score 4+ on AIME are usually missing number theory and combinatorial proof experience. Focusing preparation on these two areas — rather than continuing to drill AMC-style problems — is the faster path.

The gap between preparation and competition

The competition environment introduces factors that practice does not replicate: time pressure in an unfamiliar setting, problems that look harder than they are (because novelty triggers anxiety), and the social dynamic of sitting near students who appear to be solving problems faster. Managing these factors is a skill developed through mock competitions and through the deliberate practice of sitting with unfamiliar problems for extended periods without looking at solutions.

The minimum preparation for expecting any AIME qualification: 150 AMC 10/12 problems from the past 10 years, plus systematic coverage of the four main topic areas. This is approximately 60–80 hours of focused work.