How to Prepare for the AMC 10 and AMC 12
Published · Guide · Subject: Mathematics
The AMC 10 and AMC 12 are the entry point to the most prestigious high school mathematics competition pipeline in the United States. Most students who sit the AMC for the first time find it significantly harder than they expected. The problems require not just mathematical knowledge but a kind of creative problem-solving fluency that is not developed in standard coursework. This guide covers a practical approach to preparation.
Understand what the AMC tests
The AMC is a 30-question, 75-minute multiple-choice contest. Five answer choices are provided for each question. There is no partial credit and, since 2000, no penalty for wrong answers.
The mathematical content spans: algebra (including sequences and series, polynomials, systems of equations), geometry (plane and coordinate), number theory (divisibility, primes, modular arithmetic), counting and probability (combinations, permutations, expected value), and occasionally some trigonometry and logarithms on the AMC 12. The difficulty increases throughout the exam: problems 1–10 are accessible, 11–20 are harder, and problems 21–30 require substantial mathematical maturity.
The key insight is that these topics are not the issue for most struggling students. The issue is problem-solving fluency: the ability to see what kind of approach a problem calls for, to work efficiently without wasting time on dead ends, and to check work quickly. That fluency comes only from solving large numbers of problems.
The most important resource: past exams
The AMC has been administered since 1950 (as the AMH, then the AHSME, and since 2000 as the AMC). Decades of past problems are freely available on the Art of Problem Solving (AoPS) AMC wiki, with full community-written solutions for every problem.
The most efficient preparation strategy is to work through past exams under timed conditions, then carefully study every problem you got wrong or skipped — not just reading the solution but understanding why the solution works and what you would have needed to see to find it yourself. Solving fifty problems and fully understanding all fifty is more valuable than solving two hundred problems and skimming the solutions for the ones you missed.
Topics to fill in
After doing several past exams, most students identify gaps in specific areas. Common areas that standard coursework does not cover well:
- Modular arithmetic and number theory: Divisibility rules, prime factorization, properties of remainders, Fermat’s little theorem (AMC 12 level).
- Combinatorics: Stars and bars, overcounting and correction, the difference between combinations and permutations in context.
- Geometric probability: Area-based probability problems.
- Properties of special triangles and circles: The problems assume facility with 30-60-90 and 45-45-90 triangles, inscribed angle theorem, power of a point.
- Algebraic manipulation: Completing the square, Vieta’s formulas, clever substitutions.
The AoPS textbooks — particularly Introduction to Counting & Probability and Introduction to Number Theory — cover these topics at the appropriate depth. They are written specifically for the competition audience and are more efficient preparation than pulling from a general textbook.
Time management on test day
75 minutes for 30 problems is 2.5 minutes per problem on average. That is not enough time to attempt every problem if you are working through them in order — the last problems can take ten minutes even for strong students.
A practical strategy: work through the exam once, marking problems as “done,” “skip for now,” or “check this answer.” Complete all the problems you can do quickly. Then use remaining time on the harder problems in the middle range, where careful work can recover a correct answer. Problems 25–30 should only receive serious time if you have genuinely addressed all the earlier problems.
Because wrong answers no longer carry a penalty, blanks are slightly worse in expectation than random guessing, but the real risk is spending time on a hard problem that could be spent confirming a correct answer on an easier one.
Realistic timelines
Starting from a strong algebra/geometry background (good AMC 8 performer): Six months to a year of focused preparation, including systematic review of number theory and combinatorics, followed by regular timed practice, can produce a mid-range AMC 10 score. AIME qualification typically requires two or more years of serious preparation for most students.
Starting as an AMC 10 participant who wants to improve: Study the problem types you miss, not just the ones you get right. Keep a problem log: when you miss a problem, write a sentence about what you did not see. Review that log regularly. Consistent improvement on the AMC comes from closing specific gaps, not from more general mathematical study.
Starting as a MATHCOUNTS veteran: You already have speed and problem-solving instinct. The AMC 10/12 adds complexity in algebraic manipulation and number theory that may not have featured heavily in your MATHCOUNTS preparation. Focus specifically on those areas.
If your school does not offer the AMC
You can take the AMC at any registered site — many schools and universities register as contest sites specifically to serve students from schools that do not offer it. The MAA website allows you to search for sites in your area. If no convenient site exists, contact the MAA about options for remote or test-center-based access.
About this guide: Meli Review publishes preparation guides for academic contests alongside its contest directory. For the full structure of the AMC/AIME/USAMO pipeline, see the High School Math Olympiad Pipeline directory page.