The Syllabus, Honestly
Four pillars, each appearing on every USAMO/USAJMO: (1) ALGEBRA — inequalities (AM–GM, Cauchy–Schwarz, Schur, rearrangement, Jensen/convexity, smoothing, SOS), polynomials (Vieta, roots of unity, integer-coefficient divisibility), functional equations (substitution, Cauchy's equation, injectivity/surjectivity arguments), sequences and recurrences; (2) NUMBER THEORY — divisibility and gcd, modular arithmetic and orders, primitive roots, lifting the exponent, Vieta jumping, Diophantine equations, p-adic valuations, Zsigmondy as a citable tool; (3) COMBINATORICS — pigeonhole, extremal principle, invariants and monovariants, double counting, bijections, induction on structures, graph theory (degrees, trees, bipartite, Turán/Mantel), games and strategies, coloring arguments; (4) GEOMETRY — angle chasing, similar triangles, power of a point and radical axes, cyclic quadrilaterals, Ceva/Menelaus, homothety and spiral similarity, inversion, complex/coordinate bash as a last resort.
Six problems over two days, points each, full proofs required. Graders reward complete rigorous arguments; a correct answer without proof earns . Write-ups matter: state what you are proving, label lemmas, handle equality cases and edge cases explicitly, and never skip the "why" of a claimed step. Problems 1 and 4 are entry-level; 3 and 6 are the hardest.