USAMO Prep for Top High-School Mathletes
1-on-1 training with former IMO medalist coaches

Solving functional equations on ℝ, ℚ, and ℤ. Cauchy’s equation f(x + y) = f(x) + f(y) and its variants under continuity, monotonicity, or boundedness. Functional equations with multiple variables: f(x, y), f(x + y, z), and similar. The technique of finding fixed points and using them to constrain the function. Functional equations in number theory: equations on the integers with multiplicative structure. Proving uniqueness of solutions and writing the uniqueness argument cleanly.

Vieta’s formulas and Newton’s identities in their full form. Polynomial interpolation via Lagrange and via finite differences. Multiplicity and the derivative criterion. Resultants and discriminants of polynomials. Polynomial congruences and Hensel’s lemma at introductory level. Problems involving polynomials with integer or rational coefficients: showing certain polynomials must have integer roots, or must factor in certain ways.

AM-GM, Cauchy-Schwarz, rearrangement, Chebyshev, Jensen’s inequality with their full equality conditions. Weighted versions of the classical inequalities. The SOS (sum of squares) method, Schur’s inequality, and tangent line tricks. Hölder’s inequality and its applications. Substitution techniques: u = a + b + c, v = ab + bc + ca, w = abc normalizations. Identifying which inequality technique a problem requires from its structure.

Roots of unity and the roots-of-unity filter. Cyclotomic polynomials. Complex coordinates for geometry: representing the plane as ℂ, rotations as multiplication by e^iθ, reflections, similarity. Complex bash for proving classical theorems (Napoleon’s theorem, the Simson line, the nine-point circle). Using complex methods to prove real algebraic identities elegantly.

Linear recurrences and characteristic equations with repeated and complex roots. Generating functions (ordinary and exponential). Solving nonlinear recurrences via clever substitutions. Convergence considerations for olympiad-relevant series. Asymptotic estimates and bounding arguments — when a problem requires showing a sequence grows or shrinks at a particular rate.

Fermat’s Little Theorem, Euler’s theorem, Wilson’s theorem with full proofs and applications. The Chinese Remainder Theorem with its uses and limitations. Primitive roots, discrete logarithms, and indices. Quadratic residues, the Legendre symbol, quadratic reciprocity. Computing roots modulo p via the Tonelli-Shanks algorithm at conceptual level.

The p-adic valuation v_p and its properties. The Lifting the Exponent Lemma in full generality, with the cases for odd primes, for p = 2, and for sums versus differences. Proving non-existence results via p-adic obstructions. The Hasse principle at conceptual level: when local solvability implies global solvability and when it does not.

Linear and quadratic Diophantine equations. Pell’s equation, fundamental solutions, and the structure of all solutions. Vieta jumping in its full form: viewing a Diophantine equation as a quadratic in one variable, using Vieta’s to relate roots, applying infinite descent or extremal arguments. The classical Vieta jumping problems (IMO 1988 Problem 6 and its descendants).

Multiplicative arithmetic functions in depth. Dirichlet convolution. Möbius inversion and its applications. Computing Euler products. Recognizing problems where multiplicative decomposition simplifies the analysis. Number-theoretic generating functions.

Existence proofs via construction: building integers with prescribed properties. Non-existence proofs via modular obstructions, descent, or bounding. The interplay of additive and multiplicative structure. The technique of constructing infinitely many solutions from finitely many. Using algebraic identities to generate parameterized families of solutions.

The triangle centers and their relationships, with full proofs. The Euler line, nine-point circle, and the Euler-Feuerbach connection. Isogonal and isotomic conjugates. The Simson line, the orthic triangle, the medial triangle, and the tangential triangle. The incircle’s contact triangle. Recurring olympiad configurations and how to recognize them.

Power of a Point in its full generality. The radical axis as the locus of points with equal power. The radical center of three circles. Coaxial systems of circles and their structure. Using power-of-a-point to prove concurrence and collinearity at olympiad level.

Cyclic quadrilateral properties and Ptolemy’s theorem with proof. Ptolemy’s inequality and its equality case (when four points are concyclic). Brahmagupta’s formula and its proof. Generalized Ptolemy for non-Euclidean settings.

Rotations, reflections, translations, dilations, and spiral similarities — full classification and properties. Spiral similarities and their fixed points: the spiral similarity lemma. Composition of transformations and the resulting transformation type. Using transformations to map hard configurations to known easy ones.

Inversion in a circle with full properties. Lines and circles through the inversion center map to lines; other circles map to circles. Tangency preservation. Cross-ratio preservation. Using inversion to handle configurations with multiple tangent circles, with concyclic points, or with special points. The Apollonius problem (finding a circle tangent to three given circles) solvable via inversion.

Poles and polars with respect to a circle or conic. The cross-ratio of four collinear or concyclic points. Harmonic conjugates and harmonic ranges. Projective transformations: the projective plane, perspectivities, projectivities. The theorem that any quadrilateral can be mapped to any quadrilateral by a projective transformation. When projective methods solve a problem that synthetic and inversive methods cannot.

Coordinate bash as a problem-solving tool. Cartesian coordinates for problems with strong axis-of-symmetry structure. Complex coordinates for problems involving rotation, similarity, or concyclicity. Barycentric coordinates for triangle-centric problems: representing points relative to a reference triangle, computing concurrences and collinearities via barycentric algebra. Trigonometric identities applied to angle-chasing problems at olympiad depth.

Geometric problems with discrete or counting structure. Counting points, lines, intersections, regions. The Erdős-Szekeres theorem and its proofs. Helly’s theorem and its applications to convex sets. The Sylvester-Gallai theorem on collinear points. The Happy Ending problem. Geometric Ramsey theory at introductory level.

Bijective arguments, double counting, and inclusion-exclusion at olympiad level. Catalan numbers and their many incarnations: balanced parentheses, Dyck paths, binary trees, triangulations. Counting lattice paths via the reflection principle. Counting via generating function coefficient extraction. The exponential formula relating connected and general counts.

The pigeonhole principle with all its variants: simple, generalized, and infinite. Constructing the pigeons and holes — usually the hard part of a pigeonhole argument. The extremal principle: considering the extremal element to force structural conclusions. The well-ordering principle as the foundation. Discrete extremal arguments for combinatorial games and configurations.

Invariants under the operations allowed in a problem. Coloring arguments as invariant proofs. Constructing invariants from partial information. Monovariants: quantities that strictly increase or decrease, used to prove termination of iterative processes. The interplay between invariants (which prove non-reachability) and monovariants (which prove termination).

Graphs, trees, connectivity. Euler tours, Hamiltonian paths, planarity. Bipartite graphs and matchings. Hall’s marriage theorem, König’s theorem, and the max-flow min-cut theorem at introductory level. Ramsey theory: R(3, 3), R(4, 4) bounds, and the general statement of Ramsey’s theorem. Extremal graph theory: Turán’s theorem.

Two-player games with perfect information. Nim and the Sprague-Grundy theorem. Computing Grundy values for game positions. Strategy stealing arguments and when they apply. Pairing strategies. Games on graphs and on game trees.

Proving existence by showing positive probability over a random construction. Linearity of expectation as a proof technique. The deletion method. The Lovász local lemma at conceptual level. Concentration inequalities (Markov, Chebyshev) and their olympiad applications.

Linear algebra in combinatorics: using vector spaces over finite fields to count or to prove existence. The dimension argument: a set of vectors that spans a space cannot have too few elements. Applications to combinatorial designs, codes, and Boolean function problems.

Problems combining counting and number-theoretic structure. Sum-free sets, sumsets, and additive combinatorics at introductory level. Cauchy-Davenport theorem. The Erdős-Ginzburg-Ziv theorem. Problems involving sequences of integers with prescribed divisibility properties.