How to Prepare for Maths Olympiad — AMC, UKMT, IMO Guide
Maths Olympiad preparation is unlike any other kind of academic preparation. The problems don't test how much you know — they test how creatively you can apply a toolkit of ideas to problems you've never seen before. This guide explains how to start, what to study, and how to train effectively at every competition level.
What is competition maths?
Competition mathematics (also called olympiad maths) is a discipline that runs in parallel with school mathematics but requires completely different skills. School maths teaches you techniques and asks you to apply them in predictable ways. Competition maths gives you a problem that requires genuine insight and often multiple different mathematical ideas working together.
The problems are hard by design — many competition problems are unsolved for hours or days even by very strong mathematicians encountering them for the first time.
The competition ladder
Most countries have a hierarchy of competitions that students work through:
- Junior level: AMC 8, UKMT Junior Mathematical Challenge, SMO Junior, SASMO. Multiple choice, 25–30 questions, 40–60 minutes. Accessible to students aged 11–13
- Intermediate level: AMC 10, UKMT Intermediate Mathematical Challenge, AMC (Asian Mathematics Competition). Multiple choice with some short answer. Age 14–16
- Senior level: AMC 12, UKMT Senior Mathematical Challenge, SMO Senior. Gateway to national olympiad teams
- Olympiad level: AIME (USA), BMO1 and BMO2 (UK), USAMO, USAJMO, Senior Mathematical Olympiad (UK). Proof-based, extended problems worth full marks for correct proofs
- International: IMO, APMO, HMMT, Putnam. Elite competitions requiring years of dedicated training
The four pillars of olympiad mathematics
Almost all olympiad problems draw on four core areas. You need genuine depth in all four:
1. Number Theory: divisibility and primes, modular arithmetic, Fermat's Little Theorem, Euler's theorem, Chinese Remainder Theorem, Diophantine equations. This is the area most underrepresented in school curricula. Many students are weak here simply because they've never been taught it systematically.
2. Combinatorics: counting principles (inclusion-exclusion, stars and bars, generating functions), graph theory, the Pigeonhole Principle, probability. Combinatorics problems are often accessible from a knowledge standpoint — the difficulty is in creative application, not techniques you haven't seen.
3. Geometry: classical Euclidean geometry (similar triangles, cyclic quadrilaterals, angle chasing), trigonometric methods, coordinate geometry for specific problem types. Olympiad geometry is substantially more advanced than school geometry. You need to know a specific vocabulary of circle theorems, concurrency results and transformation arguments.
4. Algebra: inequalities (AM-GM, Cauchy-Schwarz, power mean), functional equations, polynomial theory (Vieta's formulas, factor theorem, rational roots). AMC and AIME are heavy on algebraic manipulation; later olympiads (USAMO, IMO) tend toward deeper arguments.
Resources for olympiad preparation
The best freely available resources:
- Art of Problem Solving (AoPS): artofproblemsolving.com — the single best online resource for competition maths. Past AMC, AIME and USAMO problems with full solutions. The AoPS textbooks (Introduction to Algebra, Number Theory, Counting and Probability, Geometry) are the standard preparation texts
- UKMT past papers: ukmt.org.uk — all past Junior, Intermediate and Senior Challenge papers with solutions, plus BMO papers and solutions
- Evan Chen's olympiad resources: web.evanchen.cc — detailed notes on competition mathematics from an IMO gold medallist, freely available
- Paul Zeitz — The Art and Craft of Problem Solving: the best single book for developing problem-solving thinking, not just content knowledge
How to train effectively
Competition mathematics rewards consistent, sustained practice over years — not intensive cramming before competitions. Here's an effective training structure:
- Work through topic-specific problem sets in each of the four areas before trying to solve mixed competitions. If you can't solve modular arithmetic problems, working through AMC papers won't teach you modular arithmetic — it will just reveal that you can't do it
- Spend real time on problems. A 20-minute attempt on a hard problem before looking at the solution is worth more than reading ten solutions without attempting the problems. The struggle is where learning happens
- Read solutions critically. After solving (or failing to solve) a problem, read the solution carefully. Understand not just what the solution does but why each step was chosen — what insight or idea is the key to unlocking the problem?
- Keep a problem journal. Write down problems you couldn't solve and the key insights needed for each. Review this regularly. Many olympiad insights recur across different problems
A realistic timeline
Students often ask how long it takes to reach a given competition level. Very rough estimates for a motivated student:
- AMC 8 / UKMT Junior qualifier: 3–6 months of regular practice from a strong school maths base
- AMC 10 score of 100+ (AIME qualifier): 1–2 years of consistent preparation
- AIME score of 8+ (USAMO/USAJMO consideration): 2–4 years
- USAMO / IMO qualifier: 4–7 years for most students, though exceptional students reach this faster
These are averages with enormous variance. The most important variable is how much genuine problem-solving thinking you do — not hours spent, but hours spent struggling productively with genuinely hard problems.
The role of a maths olympiad tutor
An experienced olympiad tutor can significantly accelerate development by diagnosing specific weaknesses (most students are much weaker in one of the four areas than the others), selecting problem sets at the right difficulty level (too easy builds no skills; too hard builds only frustration), and helping students understand the key ideas behind solutions they couldn't find themselves. They can also provide motivation and structure that's hard to maintain through self-study alone — particularly in the years where progress feels slow.
At The Tuition Room, our Olympiad tutors include former IMO participants, USAMO and BMO qualifiers, and national team coaches. The first session is free.