$\forall$USAMO (Olympiad Proofs)

Pigeonhole and Extremal Principles

Two ways to prove something exists without finding it. Choose pigeonholes so that sharing one forces the conclusion, and look at the largest or smallest object to make a configuration crack.

Invariants and Monovariants

Process problems ask "can you reach this state?" or "must this end?". Find a quantity that never changes, or one that only moves one way, and the answer falls out, from chameleons and chessboards to Nim.

The Essential Inequalities

AM–GM, QM–AM, Cauchy–Schwarz and their equality cases, weighted AM–GM for lopsided sums, and the art of choosing what to pair in Cauchy–Schwarz. Plus the tangent line trick for symmetric sums.

Olympiad Number Theory

The proof-level toolkit: gcds of exponents, what orders say about the prime divisors of aⁿ ± 1, Vieta jumping from the legendary IMO 1988 problem, and factoring equations into coprime pieces.

Olympiad Combinatorics

Prove something can or cannot be done, count the same set two ways, and model problems as graphs. Colourings that block tilings, incidences in the Fano plane, and Ramsey's theorem that complete disorder is impossible.

Olympiad Geometry

The configurations that recur in olympiad geometry: the incentre–excentre lemma, the radical axis and radical centre, and homotheties and spiral similarities that map one figure onto another.

Olympiad Algebra

Algebra at proof level: the divisibility a − b | P(a) − P(b) that rules out integer polynomials, functional equations solved by substitution, and roots of unity used as factor tests and filters.

Circles, Tangents and Cyclic Quadrilaterals

The circle toolkit every olympiad geometer reaches for: inscribed and tangent–chord angles, the test for four points to be concyclic, tangent lengths, and power of a point, with proofs short enough to rebuild.

Colourings, Games and Graphs

Three structures behind olympiad combinatorics: colourings that prove impossibility, games won by symmetry and invariants, and graphs that turn word problems into pictures. Proofs and worked problems for each.

Residues, Diophantine Equations and Factorials

Deeper arithmetic for olympiads: which numbers are sums of two squares, Wilson's theorem in action, how primes sit inside factorials and binomial coefficients, and how to bound a Diophantine equation into a finite check.

Polynomials, Recurrences and Radical Equations

Olympiad algebra at proof level: polynomials pinned down by their values, integer recurrences that behave, and radical equations where squaring invents fake solutions that you must catch.

The Full Inequality Ladder

Every standard inequality in its place: power means, rearrangement, Hölder, Schur and Muirhead, with the proof idea behind each and a guide to which one a problem is asking for.

Circumcircle, Incircle and Inversion

Circles doing the work: the circumradius and inradius formulas and Euler's distance between their centres, the orthocentre's reflections and the nine-point circle, and inversion, which turns circles into lines.

Exponential Diophantine Equations

Equations with unknowns in the exponents, like 2^x + 1 = 3^y. Choose a modulus where the powers cycle, factor what you can, and use Lifting the Exponent to see exactly how many times a prime divides aⁿ − bⁿ.

Functional Equations

Solve for an unknown function. Pin it down with clever substitutions, use injectivity and surjectivity as levers, and see why Cauchy's equation f(x + y) = f(x) + f(y) forces f(x) = cx on the rationals.

Bijections and Generating Functions

Count one set by matching it perfectly with another you can already count. Build and check bijections (mountain ranges, compositions, Euler's partition theorem), then let generating functions and the roots-of-unity filter do the bookkeeping.

Complex and Barycentric Coordinates

When synthetic ideas run out, bash. Put the circumcircle on the unit circle and use complex numbers, or use barycentric coordinates where the centroid, incentre and areas have one-line formulas. Plus how to choose and finish a bash.

Primitive Roots, Residues and Bounding

Primitive roots turn multiplication mod p into addition of exponents, quadratic residues decide which equations can hold mod p, and squeezing between consecutive squares kills Diophantine equations in one line.

Jensen, Rearrangement and Sequences

Convexity as an inequality machine (Jensen and Karamata), the rearrangement inequality, and integer sequences that hide a closed form. With the classic result that sin A + sin B + sin C is largest in the equilateral triangle.

Graphs, Games and Combinatorial Geometry

Count triangles by double counting, win games by monovariants, and find structure in points: lines cutting the plane into regions, the circle-cutting surprise, and the Happy Ending problem about convex quadrilaterals.

Proof Mastery

From scratch work to a clean olympiad solution. Experiment and conjecture on scratch paper, isolate lemmas so each step is checkable, and write an exposition a grader can follow at a glance.

Directed Angles and Projective Tools

Make angle chases bulletproof with directed angles mod 180°, then meet the projective toolkit: cross-ratio, Pascal and Desargues, and named configurations like the Simson line and the Miquel point.

Set Systems and the Probabilistic Method

Extremal problems about graphs and families of sets: how many edges force connectivity or planarity to fail, Sperner's and Erdős–Ko–Rado's theorems on set families, and proving existence by showing a random object works with positive probability.

Legendre, Pell and Möbius

Three classical tools: Legendre's formula through digit sums, Pell equations and their infinitely many solutions from one, and Möbius inversion, which untangles sums over divisors.

Symmetric Functions and the Pointwise Trap

Two last algebra lessons. Power sums and elementary symmetric functions convert into each other through Newton's identities. And the classic functional-equation trap: proving f(x) = x or f(x) = −x at each point is not the same as for all x at once.