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Twelve exercises: your first proof practice plan · Try the mixed readiness check · Look up unfamiliar terms
| 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.
A productive study session
Begin with a problem you cannot solve immediately. Write down what is given and what must be proved. Try small cases, draw a diagram where useful, and look for a simpler related problem. Record your attempts before reading a hint.
After studying a solution, close it and reconstruct the argument yourself. Later, revisit a related problem to see whether you can recognise and adapt the idea.
Balance four kinds of work
- Learn one concept or technique carefully.
- Solve unfamiliar problems without immediate hints.
- Write complete solutions, including easy-looking steps that carry the reasoning.
- Review old work and explain why unsuccessful approaches failed.
Keep a problem journal
Record the source, your attempt, the useful idea, and the gap in your original reasoning. Organise it by methods as well as subjects: pigeonhole, extremal choice, useful substitution, auxiliary construction or modular obstruction.
Choose the right difficulty
A national preliminary paper and an IMO paper serve different purposes. Work first at the level of your next competition. Move to harder problems when you can produce dependable complete solutions, not merely recognise a technique after seeing it.
Study the syllabus
Choose the concepts you need next.
Explore guide PRACTISEOfficial resources
Find released papers and training materials.
Explore guideChoose your first paper
Open the country paper directory and select the stage you are entering. Read the instructions before timing yourself. If a proof paper feels inaccessible, return to the prerequisite topic and solve a shorter problem using the same idea. Keep one complete written solution to compare with your later work.
Begin with a worked lesson
| Readiness | Try this lesson | What to check |
|---|---|---|
| Build foundations | Fractions, equations and parity | Can you explain every step? |
| Bridge to proofs | Induction and counterexamples | Have you covered every case? |
| Develop a method | Remainders, gcd, power cycles and integer equations: four examples and nine exercises | Can you explain why the method works? |
| Write a geometry proof | Congruence with a labelled diagram | Which hypothesis justifies each inference? |