The main fields are algebra, number theory, geometry and combinatorics. This is a preparation curriculum: the IMO does not issue an exhaustive chapter-by-chapter syllabus. National first rounds may have narrower topic lists. Our topic map is an editorial study guide, not a regulation issued for every country. The IMO 2027 organiser’s overview describes the international problem areas and the role of elementary mathematics.
Algebra
Patterns, equations and the art of a useful transformation.
Explore guide 02Number theory
Remainders, gcd and integer equations: four worked examples and nine exercises with full solutions.
Explore guide 03Geometry
Turn a diagram into a chain of precise ideas.
Explore guide 04Combinatorics
Count, construct and discover what must be true.
Explore guideFoundations
School mathematics, notation and logical language.
Explore guide THINK CLEARLYProof-writing
Direct proof, contradiction, induction and construction.
Explore guideHow much advanced mathematics do you need?
Start with elementary ideas and learn to apply them deeply. Calculus is not a central preparation strand. Advanced tools such as inversion, generating functions and primitive roots are selective extensions, not a beginner checklist. There are no fixed topic percentages guaranteed for an IMO paper.
A sensible learning progression
| Phase | Main work | Readiness checkpoint |
|---|---|---|
| A. Build foundations | Fractions, identities, equations, divisibility, angles, counting | Solve unfamiliar short problems and explain the main reason |
| B. Bridge to Olympiads | Gcd, congruences, similarity, circles, basic inequalities, pigeonhole | Write short complete proofs in each subject |
| C. Build national-round strength | Polynomials, Diophantine equations, concurrency, invariants, double counting | Sustain a multi-step argument and repair a flawed proof |
| D. Prepare for selection tests | Hard mixed sets, functional equations, transformations, advanced combinatorics | Decide which problems to pursue and communicate partial progress clearly |
| E. Practise IMO format | Released past papers under timed conditions, then detailed review | Produce correct, readable solutions under contest constraints |
Progress by demonstrated skill, rather than promising a fixed number of months. Younger learners can begin with enrichment; the relevant country’s rules determine when they may enter a selection competition.
Choose the specification for your next round
The first round may test school mathematics or short answers; later rounds usually demand proofs. Use the stage, division and season on your national notice. For examples, compare American AMC information, Japan’s JMO overview, Taiwan’s selection office and India’s HBCSE syllabus. These govern different national contexts.