IMO (International Mathematical Olympiad) Exam Survival Guide

Window: Proof-based Contest | Areas: Algebra, Number Theory, Combinatorics, Geometry

Must-do topics

• NT: divisibility, orders, LTE basics, lifting exponents, bounding, invariants
• Algebra: inequalities (AM-GM, Cauchy, Jensen), functional equations, polynomials (Vieta, root bounds)
• Combinatorics: Pigeonhole, double counting, constructive/existence, invariants/monovariants
• Euclidean Geometry: angle/length chasing, cyclic/quadrilateral power of a point, homothety, similarity, bary/vec hints

Top traps (avoid)

• Proving “if” but not “only if” (half-proofs)
• Missing edge cases (0, 1, negatives, extremals)
• Hand-wavy diagram arguments without lemmas
• Algebra bash without identifying equality cases

Study split
40% problem sets (past IMOs/shortlists) | 30% theory lemmas | 20% write-ups | 10% timed mocks

Last-48h checklist

• Re-solve 6 past Q2/Q3s (sketch → full write-up)
• Geometry: compile your 12 go-to lemmas (PoP, Miquel, Ceva, Menelaus, etc.)
• Inequalities: equality cases + standard substitutions (t = x/y, homogenization)
• Sleep routine + paper layout rehearsal

Quick facts/frames

• When stuck: try invariants/extremal element; consider parity; look for symmetry
• FEQs: test constant/linear; plug x=0,1,−x; swap (x,y)
• Inequalities: homogenize → apply AM-GM/Cauchy → check equality

Speed tactics

• Spend 15–20 min mapping structure (claims/lemmata) before writing
• Draw big, mark equal angles/parallel lines early
• If algebra explodes, search for a hidden substitution

Exam-day mini-plan
Q1 warm-up (≤45 min) → map Q2/Q3, pick one to push → leave 30–40 min for clean write-up and edge-case audit