Algebra & Inequalities
AM–GM (weighted too); Cauchy–Schwarz and Engel; QM–AM–GM–HM; power means; Jensen; rearrangement and Chebyshev; Schur; SOS; Hölder ; Muirhead for symmetric sums; tangent-line trick; smoothing.
Locate the equality case; homogenize; choose the inequality whose equality case matches; finish with SOS if stuck. Write the equality condition explicitly in the proof.
Vieta and Newton; ; rational root theorem; Eisenstein and mod- irreducibility; Lagrange interpolation and finite differences; roots of unity filter; Chebyshev polynomials for boundedness. FE: substitutions (, , , , ), Cauchy's equation on and with regularity on , injectivity/surjectivity lemmas, fixed points, subtracting a particular solution.
Verify every FE candidate in the original equation. For polynomials, compare degrees and leading coefficients, count roots, and use integer-value constraints.
Number Theory & Combinatorics
Orders, primitive roots, CRT; Fermat, Euler, Wilson; LTE (all cases); Legendre, Kummer; quadratic residues, Euler's criterion, reciprocity and both supplements; Vieta jumping; infinite descent; Zsigmondy (citable); ; sums over divisors and Möbius inversion; -adic valuation bookkeeping; bounding between consecutive powers.
Reduce mod first; then valuations; then orders. For "", the order of must be even. For symmetric quadratic Diophantines, Vieta jump from a minimal solution.
Pigeonhole (all forms); extremal principle; invariants and monovariants; colorings and weightings; double counting; bijections; induction on structures; PIE; generating functions; Catalan and reflection; Erdős–Szekeres; graph theory (degrees, trees, bipartite, Mantel/Turán, Hall, Euler's formula); games (backward induction, P/N positions, Nim and XOR, symmetry and strategy stealing).
Ask: what never changes (invariant)? what only decreases (monovariant)? which object is extreme? which two things can I count? If the problem is a game, build the P/N table backward and prove the pattern.
Geometry
Angle chasing and cyclic quadrilaterals (opposite angles, equal inscribed angles, the exterior-angle criterion); power of a point and radical axes/center; Ptolemy (and its inequality); Brahmagupta; Ceva and Menelaus (trig forms too); Stewart, Apollonius, angle-bisector length; incircle tangent lengths ; Euler line and nine-point circle; homothety; spiral similarity and Miquel points; inversion; the British flag theorem; complex numbers on the unit circle; barycentric coordinates for cevian-heavy problems.
Find the circle. Then: inscribed angles, power of a point, Ptolemy — in that order. Concurrency: Ceva (trig form for angles). Collinearity: Menelaus or a well-chosen homothety. Bash only when the configuration is tame, and set up coordinates to exploit symmetry.
Restate the claim; fix notation; prove lemmas before use; name every theorem; cover every case (including degenerate ones); state the equality condition; conclude.
A USAMO solution is a piece of technical writing. Numbered steps, one idea per paragraph, every "clearly" replaced by a reason. After writing, read it as a hostile grader looking for the step you skipped.