USAMO (Olympiad Proofs)

How to structure a rigorous argument

Existence from counting and extremes

Quantities that never change, or move one way

AM–GM, Cauchy–Schwarz, smoothing

Quantities that never change, elements that must exist

The four pillars and the complete checklist

Orders, LTE, Vieta jumping, and valuations

Invariants, extremal arguments, double counting, graphs

Power of a point, radical axes, spiral similarity, inversion

Inequalities at proof level, polynomials, functional equations

Tangent lengths, Brahmagupta, Ptolemy, the British flag, and where circles hide

Extremal boards, matchings, no-adjacency counts, forests and cycles

Quadratic residues, difference-of-squares counts, Legendre gaps, Euler at work

Vieta in reverse, Newton in miniature, characteristic roots, extraneous solutions

AM–GM with constraints, Engel form, Cauchy as geometry, QM–AM, SOS and Schur

Arc midpoints, the incenter–excenter lemma, power of a point, Euler's OI², bisector lengths

Powers of two, difference of squares, Catalan-type equations, and the parity–valuation reflex

Substitutions that isolate f, injectivity from the equation, monotone f(f(n)) = 3n, Möbius orders, d'Alembert

Reflection, rooks and derangements, Fibonacci diagonals, matchings by inclusion–exclusion, homothety in figures

Timed sets from the USAMO pool, and how to write proofs that score

The USAMO reference: every tool, with the proof-level method that wields it