AIME Problems & Practice Tests: Free 2026 AMC Prep Guide

The AIME (American Invitational Mathematics Examination) is a challenging 15-question math competition exam administered annually to high school students who qualify through the AMC 10 or AMC 12. Unlike the multiple-choice AMC exams, AIME answers are integers between 000 and 999 — no answer choices are provided. Every question requires constructing a complete solution from scratch, which demands both deep mathematical knowledge and creative problem-solving under time pressure.

AIME is the gateway to the USA(J)MO (USA Mathematical Olympiad), USAMO/USAJMO — the pathway to the International Mathematical Olympiad (IMO) team. But even for students not targeting the Olympiad pathway, a strong AIME score distinguishes college applicants at selective institutions, signals exceptional mathematical ability to university mathematics departments, and provides preparation for the analytical thinking demanded by rigorous STEM programs.

The exam is divided into two separate offerings: AIME I (typically administered in February) and AIME II (typically administered in March). Qualifying students may take either or both exams. The stronger score is submitted for USAMO/USAJMO qualification purposes. Both exams cover the same content domains but feature independently constructed problem sets — preparing for both through extensive practice across all content areas is the most effective strategy.

The difficulty progression within each AIME is steep. Problems 1–5 are generally accessible to prepared students; problems 6–10 require creative problem application; problems 11–15 challenge even the most experienced competition mathematicians. Most students who qualify for AIME score between 1 and 7 on their first attempt. An AIME score of 10 or above places a student in the top tier of competition mathematics nationally, and scores of 12–15 are typically required for USAMO/USAJMO qualification depending on the year.

Effective AIME preparation requires a different approach than AMC preparation. The AMC rewards speed and elimination strategy — the AIME rewards depth, persistence, and creative mathematical insight. Students who perform well on AMC through pattern recognition and test-taking strategy often find their first AIME more challenging than expected, because the exam demands genuine mathematical understanding rather than familiarity with common problem types.

The practice tests here are organized by mathematical topic, building the domain-specific depth that AIME success requires. Consistent practice across all AIME topic areas — not just the topics where you're already comfortable — is what separates students who plateau at 4–6 from those who reach 8+ consistently. Uncomfortable practice is productive practice: the problems that challenge you most are the ones that build the most durable mathematical skill over time.

The AIME's format distinguishes it from virtually every other standardized exam a student encounters. There are no answer choices — each of the 15 problems requires computing an exact integer answer between 000 and 999. This format has two important implications: you cannot guess strategically, and the full solution must be derived correctly. Partial credit is not available — an answer of 358 earns no points even if the correct answer is 359.

The 15 questions are presented in rough order of difficulty, though the actual difficulty curve varies by student depending on mathematical strengths. Most students find problems 1–5 accessible with AIME preparation, problems 6–10 challenging, and problems 11–15 genuinely hard. The optimal test-taking strategy for most students is to work through the first 10 problems carefully, skip problems where you're stuck after a few minutes, then return to skipped problems in the remaining time.

Time management is critical but often mishandled by first-time AIME takers. The 3-hour limit (12 minutes per problem on average) is generous for early problems but inadequate for later ones without strategic allocation. Experienced AIME competitors often spend 5–8 minutes on problems 1–5, 10–15 minutes on problems 6–10, and 20+ minutes on problems 11–15. Knowing when to move on is a learned skill — practicing with timed sessions before the exam builds the awareness of when a problem isn't yielding to your current approach.

Computation accuracy is a significant differentiator on the AIME. Because answers must be exact integers and computation is entirely by hand, arithmetic errors in complex multi-step solutions can cost correct answers despite correct problem-solving approaches. Systematic habits — organizing work clearly, checking partial computations, verifying the answer falls in the 000–999 range — reduce costly errors. Practice computing complex expressions without a calculator, developing facility with modular arithmetic for large numbers and careful algebraic manipulation for multi-variable expressions.

One underappreciated aspect of AIME preparation is familiarity with the answer format itself. Because answers must be integers between 000 and 999, knowing this constraint allows you to use it as a check throughout your solution. If your algebra yields a non-integer or a value outside this range, you've made an error somewhere — or you're interpreting the question incorrectly.

Competition veterans often sanity-check intermediate results against this range as they work, catching errors before they compound through multiple steps. This habit takes no extra time and significantly reduces the rate of computational mistakes near the end of a complex solution.

The absence of a calculator on the AIME is intentional — the exam is designed to test mathematical reasoning, not computational speed with a device. However, this means your mental arithmetic and hand computation must be reliable. Practice doing modular arithmetic by hand with large numbers, simplifying complex fractions, and expanding polynomial expressions without errors. Students who struggle with calculation speed often benefit from targeted practice on arithmetic fluency separate from problem-solving practice — developing reliable hand computation independently, then applying it within problems. The AIME qualification requirements outline the full scoring and cutoff structure in detail.

Effective AIME preparation requires honest assessment of your current competency across all five major topic areas — algebra, geometry, number theory, combinatorics, and advanced topics (polynomials, sequences, trigonometry/complex numbers). The most productive preparation focuses study time on the specific domains where your problem-solving breaks down, not on reviewing content you already handle well.

Algebra and combinatorics are the two areas where most students can make the most rapid score improvement with targeted preparation. Algebra problems appear frequently across the difficulty range — including in the early problems where they provide accessible scoring opportunities — and the core tools (Vieta's formulas, manipulation of symmetric functions, algebraic identities) are learnable with focused practice. Combinatorics is similarly high-leverage because the foundational tools (inclusion-exclusion, bijections, generating functions) open a wide range of problem types once they're genuinely understood rather than superficially known.

Number theory is often underestimated by students who haven't spent time with modular arithmetic. Once you're comfortable with congruences, Euler's totient theorem, and the Chinese Remainder Theorem, a significant portion of AIME number theory problems become much more tractable. The investment pays off quickly — number theory problems tend to appear in the accessible mid-range (problems 4–9) and offer clean, verifiable integer answers that confirm whether your approach is correct.

Geometry requires the deepest investment in new tools for students whose background is primarily coordinate geometry and basic Euclidean proofs. Learning Ptolemy's theorem, power of a point, mass point geometry, and Stewart's theorem opens access to many AIME geometry problems that would otherwise require unwieldy coordinate approaches. Coordinate geometry remains useful — particularly when a problem has a natural rectangular symmetry — but classical geometry tools are typically more elegant and less error-prone for AIME problems.

Advanced topics — polynomials, sequences and series, and trigonometry/complex numbers — are highest-priority for students targeting scores of 8 or above. Problems 11–15 frequently require these tools, and they appear occasionally in the 7–10 range as well. For students currently scoring below 6, mastering the foundations (algebra, geometry, number theory, combinatorics) will produce more score improvement per hour of study than reaching into advanced topics prematurely. A solid foundation in four core areas is worth far more than shallow familiarity across all five.

AIME problem-solving strategy is distinct from AMC strategy in one fundamental way: you must construct a complete solution, not recognize it from options. This means developing genuine mathematical insight rather than test-taking technique. The strategies below are drawn from how experienced competition mathematicians approach novel problems — not test-taking tricks, but genuine mathematical practices.

Start by fully understanding the problem before attempting a solution. AIME problems are precisely stated, and every condition matters. Read the problem twice, identify what you're given and what you're asked to find, and make note of constraints that might seem incidental but are actually central (divisibility conditions, integer requirements, geometric configurations). Many errors come from misreading the problem rather than from mathematical mistakes.

Work forward from what you know and backward from the answer. If you know the answer must be an integer, and your expression gives a fraction, either you've made an error or there's a simplification you haven't found. Use the answer range (000–999) as a check on your work — if your computation gives 1,847, something is wrong. Sometimes working backward from the integer constraint guides you toward the right approach.

Draw on multiple mathematical domains simultaneously. The hardest AIME problems require connecting insights from two or three different mathematical areas — a geometry problem that requires modular arithmetic, a combinatorics problem that requires algebraic manipulation. When a single-domain approach stalls, ask whether a different mathematical framework could simplify the problem. Algebraic reformulation of geometric problems, and geometric interpretation of algebraic problems, are particularly common crossover approaches.

Review your solutions thoroughly, even for problems you solved. Understanding why your approach worked — and what alternative approaches might also have worked — builds the mathematical vocabulary that transfers to novel problems. Competition mathematics is learned by accumulating problem-solving patterns: the more patterns you've internalized, the more solutions you can recognize and construct on novel problems. The AIME exam overview covers the broader competitive mathematics pathway and how the AIME fits into the progression toward olympiad-level competition.

Mental flexibility under pressure is as important as mathematical knowledge on exam day. Students who have practiced recognizing when an approach isn't working — and pivoting to a different mathematical framework — perform significantly better on harder AIME problems than students with equivalent content knowledge but rigid problem-solving habits. When you're stuck, explicitly ask: could this algebra problem be approached geometrically? Could this counting problem be reframed as a polynomial? Active strategy switching, rather than pushing harder on a failing approach, is the skill that separates scores of 5 from scores of 9.

Realistic score improvement on the AIME happens over months, not weeks. Students who go from AIME 2–3 to AIME 7–8 typically do so over 6–12 months of consistent practice; improvement from 8 to 12 requires another year or more at the frontier of competition mathematics. Understanding this timeline doesn't diminish the goal — it means that starting preparation early and maintaining consistent practice produces results that last-minute cramming cannot.

The most effective practice cadence for AIME preparation combines topic-specific practice (to build domain competency) with full timed exams (to build integration and time management). Early in your preparation, spend more time on topic-specific practice. As your exam approaches, shift progressively toward full exam simulation to build the fluency and pacing required for 3-hour performance under pressure.

Track your performance by topic across practice sessions. Knowing that you're at 4/5 on algebra problems but 1/5 on geometry guides your next week's study more precisely than a single overall score. The practice tests on this site are organized by topic precisely to support this kind of targeted development — use the topic-specific tests to build domain competency, then use the mixed and full practice exams to integrate your preparation.

Solution review is the step most students skip and the one that produces the most learning per hour. After every practice problem — especially the ones you got wrong — spend time understanding the official or community solution, including why alternative approaches work or fail.

Many AIME solutions can be solved through multiple distinct methods: a combinatorics problem might yield to both direct counting and generating functions, and a geometry problem might work through both coordinates and classical tools. Understanding multiple solution paths for the same problem builds mathematical flexibility and expands the approaches you can draw on when facing novel configurations.

Building a personal problem log is a practice used by high-scoring competition veterans. When you encounter a problem that uses a technique you haven't seen before, or that reveals a blind spot in your understanding, record it: write the key insight, the mathematical tool that unlocked it, and a similar problem type to revisit later. Over months of preparation, this log becomes a curated reference of your own learning — more valuable than any textbook because it's calibrated to your specific gaps. Reviewing it in the week before your exam consolidates the patterns you've worked to internalize throughout your preparation.

Consistency beats intensity in AIME preparation. Two focused hours of problem-solving practice five days per week produces more long-term improvement than eight-hour marathon sessions on weekends. Mathematical pattern recognition — the core skill the AIME tests — develops through regular, repeated exposure over time.

Students who study consistently across months build genuine mathematical intuition; students who cram before the exam build familiarity with specific problem types but lack the underlying flexibility the harder AIME problems require. Set a consistent weekly schedule, protect that time, and trust that the cumulative effect of sustained practice will show up in your score on exam day.