USAMO (Olympiad Proofs)
Proof
Proof-based olympiad mathematics for USAJMO / USAMO: techniques, inequalities, number theory, combinatorics, geometry and functional equations, with written-proof practice.
What you'll be able to do
- Write complete, rigorous proofs
- Use the classic olympiad techniques across all four areas
Start the course
Go through each topic step by step — prose, worked examples, quick checks, and a timed practice round after every section.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics
Pigeonhole & Extremal
3 lessonsExistence from counting and extremes
Invariants & Monovariants
3 lessonsQuantities that never change, or move one way
Inequalities
4 lessonsAM–GM, Cauchy–Schwarz, smoothing
Olympiad Number Theory
4 lessonsOrders, LTE, Vieta jumping, and valuations
Olympiad Combinatorics
3 lessonsInvariants, extremal arguments, double counting, graphs
Olympiad Geometry
3 lessonsPower of a point, radical axes, spiral similarity, inversion
Olympiad Algebra & Functional Equations
3 lessonsInequalities at proof level, polynomials, functional equations
Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals
3 lessonsTangent lengths, Brahmagupta, Ptolemy, the British flag, and where circles hide
Olympiad Combinatorics II: Colorings, Games & Graphs
3 lessonsExtremal boards, matchings, no-adjacency counts, forests and cycles
Olympiad Number Theory II: Residues, Diophantine Equations & Factorials
3 lessonsQuadratic residues, difference-of-squares counts, Legendre gaps, Euler at work
Olympiad Algebra II: Polynomials, Recurrences & Radical Equations
3 lessonsVieta in reverse, Newton in miniature, characteristic roots, extraneous solutions
Olympiad Inequalities II: The Full Ladder
3 lessonsAM–GM with constraints, Engel form, Cauchy as geometry, QM–AM, SOS and Schur
Olympiad Geometry III: Circumcircle, Power of a Point, Incircle
3 lessonsArc midpoints, the incenter–excenter lemma, power of a point, Euler's OI², bisector lengths
Olympiad Number Theory III: Exponential Diophantine Equations
3 lessonsPowers of two, difference of squares, Catalan-type equations, and the parity–valuation reflex
Functional Equations With Tricky Conditions
3 lessonsSubstitutions that isolate f, injectivity from the equation, monotone f(f(n)) = 3n, Möbius orders, d'Alembert
Bijections, Homothety & Structured Counting
3 lessonsReflection, rooks and derangements, Fibonacci diagonals, matchings by inclusion–exclusion, homothety in figures
Olympiad Geometry IV: Barycentric & Complex Coordinates
3 lessonsWhen synthetic geometry stalls: the two algebraic systems that finish olympiad problems
Olympiad Number Theory IV: Primitive Roots, Residues & Bounding
4 lessonsOrders that control exponents, quadratic residues that kill equations, and size arguments that finish them
Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences
4 lessonsConvexity and ordering as inequality engines; polynomials and sequences at proof level
Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry
3 lessonsThe four combinatorial toolkits the USAMO draws from most often
Proof Mastery: Exposition, Lemmas & Scratchpad Strategy
3 lessonsHow a 7 becomes a full-score proof: structure, lemmas, and what to do with the first ninety minutes
Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations
4 lessonsAngles mod 180°, cross ratio and polars, Simson, symmedians, isogonal conjugates, mixtilinear incircles — and when to bash
Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method
3 lessonsConnectivity and trees, Turán and Mantel, planarity, Hall, Sperner and LYM, Helly, and proving existence by averaging
Olympiad Number Theory V: Legendre, Pell & Möbius
3 lessonsValuations in factorials, the Legendre symbol, Pell equations, multiplicative functions and Möbius inversion, Bertrand
Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap
3 lessonsElementary symmetric polynomials, Newton's identities, interpolation with a forced zero, and the commonest functional-equation error