The International Mathematical Olympiad (IMO) is the highest-profile international proof-based Mathematics competition for national teams of school students. It is not an open-entry contest: students reach it through national selection and training systems.
The 67th IMO was held in Shanghai, China, from 10 to 21 July 2026. The official IMO record lists 117 countries and 666 contestants for the 2026 edition.
What IMO Mathematics looks like
IMO problems are proof problems. The core domains are traditionally algebra, combinatorics, geometry and number theory, but the real demand is deeper: invent a route, justify every step, manage cases and communicate a complete argument under time pressure.
The route is national before it is international
A Singapore student does not prepare for IMO by simply downloading IMO papers at an early stage. The practical route passes through strong local olympiad performance, selection and national training. BTT therefore links IMO downward to SMO and APMO.
Preparation principles
- Build deep domain knowledge instead of collecting tricks.
- Attempt hard problems for long enough to generate original structure.
- Write solutions completely and submit them to critique.
- Study alternative solutions for transferable ideas.
- Use past IMO problems selectively; difficulty should stretch rather than merely defeat.
The role of proof
At IMO level, proof is not presentation added after solving. Proof is the solution. Examples, diagrams and numerical experiments may reveal structure, but the final argument must establish why the claim holds in every required case.
Official source checked 26 September 2026: IMO official 2026 edition record.
World Mathematics route: return to the World Mathematics Atlas for the wider map across competitions, school Mathematics, examinations, mathematical objects and university routes.
International Mathematical Olympiad: understand the proof standard and national-selection pathway
IMO participation is through national teams
Students do not register individually for the IMO. Each participating country selects a team through its own process.
The competition is proof-based
Problems demand complete arguments in algebra, combinatorics, geometry and number theory. Calculators and routine formula application are not the core challenge.
National selection comes first
A Singapore student should follow current Singapore Mathematical Society/national selection information rather than an overseas pathway.
Domain depth must become flexible
Advanced facts are useful only when students can recognise when they apply. Training should emphasise problem structure and proof construction.
Geometry requires creative construction
The decisive line, transformation or circle may not be given. Students need a large repertoire plus the ability to invent.
Number theory needs local-to-global reasoning
Congruences, valuations, Diophantine equations and prime factor structure support impossibility and classification proofs.
Combinatorics often needs invariants or extremal principles
Counting formulas alone rarely cover the full domain.
Algebra includes inequalities and functional equations
Students need condition awareness, equality cases and complete classification of solutions.
Six problems do not mean six equal attempts
At elite level, selection of promising problems and sustained work over long periods are part of performance.
Solutions should be written for verification
Every nontrivial assertion needs a reason. Training should include rewriting proofs after feedback.
Progression takes years
Strong school Mathematics → national olympiad domains → proof competitions → national selection is a more realistic route than immediate IMO-specific drilling.
The educational value exceeds selection
Deep olympiad training can develop proof, persistence and abstraction even for students who never reach an IMO team.