IMO Coaching for Olympiad Mathletes
1-on-1 with coaches who’ve been there

Cauchy functional equations over ℝ, ℚ, and ℤ with their full classification under various regularity conditions. Pexider equations and their reduction to Cauchy. Functional equations with multiple variables: f(x, y), f(x + y, z), and their analysis. Functional inequalities and their relationship to equations. Boundedness and monotonicity as tools for forcing solutions. The technique of fixed points: finding x with f(x) = x and using the fixed point to constrain f globally. IMO problems where functional equation techniques apply across topic boundaries.

Vieta’s formulas and Newton’s identities in their full generality. Polynomial interpolation via Lagrange, Newton’s forward differences, and the divided differences formula. Resultants and the relationship between polynomial roots. Discriminants of polynomials of arbitrary degree. The fundamental theorem of algebra. Algebraic versus transcendental: when an IMO problem’s structure forces a polynomial answer to be algebraic. Hensel’s lemma at the level needed for olympiad problems.

AM-GM, Cauchy-Schwarz, rearrangement, Chebyshev, Jensen at full depth with all variants. Hölder’s inequality and its generalizations. Minkowski’s inequality. Power mean inequalities. The SOS method, tangent line techniques, Schur’s inequality. Identifying equality cases — at IMO level, problems often specifically test whether the equality case is correctly characterized. The substitution methods: a = x + y, b = y + z, c = z + x for triangle inequalities; u = a + b + c, v = ab + bc + ca, w = abc for symmetric polynomials in three variables. Recognizing when an inequality problem reduces to a single classical inequality versus requiring a multi-step argument.

Roots of unity and the cyclotomic polynomials Φ_n(x). Properties of Φ_n: degree φ(n), irreducibility over ℚ, the relationship Π_d|n Φ_d(x) = xⁿ − 1. Using cyclotomic polynomial structure in number theory and combinatorics. Complex bash for geometry: representing complex points, lines, circles in ℂ. Proving classical theorems (Napoleon’s theorem, the nine-point circle, the Simson line) via complex methods. The Lindemann-Weierstrass theorem at conceptual level — used in problems where transcendence considerations matter.

Linear recurrences with the characteristic equation in full generality, including repeated and complex roots. Nonlinear recurrences and their analysis. Ordinary and exponential generating functions. The Lagrange inversion formula at conceptual level for advanced generating function problems. Asymptotic estimates: when a problem requires bounding a sequence’s growth rate, decay rate, or behavior over many terms.

The framework of Galois theory: field extensions, automorphisms, the Galois group of a polynomial. Why certain constructions are impossible (trisecting an angle, doubling the cube, constructing a regular 7-gon). The conceptual framework matters for problems where the answer hinges on solvability properties — even without computing Galois groups directly, recognizing the underlying obstruction is the insight.

Fermat’s Little Theorem, Euler’s theorem, Wilson’s theorem at full depth. The Chinese Remainder Theorem with its uses and the limits of its application. Primitive roots and the structure of (ℤ/pℤ)*. Discrete logarithms and indices. Quadratic residues and the Legendre symbol. Quadratic reciprocity with proof sketch and applications. Higher reciprocity at conceptual level.

The p-adic valuation v_p with its full algebraic structure. Lifting the Exponent in complete generality, with all cases (odd p, p = 2, sums, differences). The Hasse principle: when local solvability implies global solvability for Diophantine equations. Newton polygons for analyzing polynomial roots p-adically. p-adic numbers at conceptual level — for problems where local obstructions matter.

Linear and quadratic Diophantine equations and their full theory. Pell’s equation with the continued fraction approach to solutions. Vieta jumping in full form, including classical problems from IMO history (IMO 1988 Problem 6 being the canonical example). The method of infinite descent. Fermat’s method of factoring. Elliptic curves at conceptual level — IMO problems occasionally hinge on properties of elliptic curves without requiring the full theory.

Multiplicative arithmetic functions and Dirichlet convolution. Möbius inversion and its applications. Euler products. The relationship between multiplicative functions and Dirichlet series. Computing sums over divisors via multiplicative decomposition. The Mertens function and prime-counting estimates at conceptual level.

Algebraic integers in quadratic fields ℤ[√d] and ℤ[ω] (Eisenstein integers). Unique factorization in these rings and the failure of unique factorization in general. Class numbers at conceptual level. Using quadratic field arithmetic to solve Diophantine problems that resist standard methods — for example, representing integers as sums of squares.

Construction of integers with prescribed properties. Non-existence proofs via modular obstructions, descent, or growth bounds. The interplay of additive and multiplicative structure in IMO problems. Parameterized families of solutions: constructing infinite families from finite cases. The technique of “smoothing”: modifying a candidate construction to satisfy additional constraints.

The triangle centers and the full network of their relationships. The Euler line, the nine-point circle, the Feuerbach point and the Feuerbach circles. Isogonal and isotomic conjugates, isogonal lines, isogonal points. The Simson line and its generalizations. The Wallace-Bolyai-Gerwien theorem. Recurring olympiad configurations: the incircle-contact triangle, the medial triangle, the orthic triangle, the tangential triangle, the antimedial triangle, the pedal triangle of an arbitrary point.

Power of a Point in full algebraic form. The radical axis as a locus and its alternative characterizations. Radical centers. Coaxial systems of circles: orthogonal coaxial systems, pencils of circles through two points, pencils tangent to a line. Using coaxial structure in problems involving multiple circles.

Cyclic quadrilateral properties at full depth. Ptolemy’s theorem and inequality with proof. Brahmagupta’s formula for cyclic quadrilateral area, with proof and applications. Generalizations of Ptolemy: Casey’s theorem for tangent circles. The relationship between cyclic quadrilaterals and the cross-ratio.

The full classification of plane isometries (translations, rotations, reflections, glide reflections) and similarities. Spiral similarities and their fixed points: the spiral similarity lemma in its full form. Compositions and the resulting transformation type. Möbius transformations as the most general angle-preserving maps in the plane (under inversion). Using transformations to map any configuration to a canonical one.

Inversion in a circle with full properties: line-to-line, line-to-circle, and circle-to-circle correspondences. Tangency preservation. Cross-ratio preservation. Inversion at a center on a circle. Computing the inverse of common configurations: triangles, cyclic quadrilaterals, tangent circles. The Apollonius problem solved via inversion. When inversion makes a problem trivial; when it requires careful choice of center and radius.

Poles, polars, and the polar duality with respect to a conic. The cross-ratio of four collinear points or four concurrent lines or four concyclic points. Harmonic conjugates and harmonic ranges. Projective transformations: the projective plane, perspectivities, projectivities, and the fundamental theorem of projective geometry. The duality principle: every projective theorem has a dual statement. Pascal’s theorem and Brianchon’s theorem. When projective methods solve problems that synthetic and inversive methods cannot.

The full toolkit of coordinate methods. Cartesian coordinates for problems with axis-of-symmetry structure. Complex coordinates for problems involving rotations, similarities, or concyclicity. Barycentric coordinates for triangle-centric problems, including normalized barycentric and homogeneous barycentric. Areal coordinates and their relationship to barycentric. Trigonometric methods at olympiad depth: identities of arbitrary complexity, applied to angle-chasing problems where pure synthetic methods become infeasible.

Counting problems with geometric structure. Lattice points, lines, intersections, regions in the plane. The Erdős-Szekeres theorem and its proof via Ramsey theory. Helly’s theorem and its applications to convex sets. The Sylvester-Gallai theorem on points not in general position. Combinatorial geometry over finite fields. Geometric Ramsey theory: the Hales-Jewett theorem at conceptual level.

Bijective arguments, double counting, inclusion-exclusion at olympiad depth. Catalan numbers and their many olympiad-relevant identities. Counting lattice paths via the reflection principle and the cycle lemma. Counting via algebraic methods: coefficient extraction from generating functions. The exponential formula. Counting with constraints: the principle of inclusion-exclusion applied to complex set systems.

The pigeonhole principle in all its variants. Constructing the pigeons and the holes — typically the hard step in IMO pigeonhole problems. The extremal principle and the well-ordering principle as its foundation. Discrete extremal arguments. The probabilistic method as a generalization of pigeonhole. Recognizing when an existence problem is solvable by counting versus when it requires construction.

Invariants under operations allowed in a problem. Coloring arguments as invariants (mod 2, mod 3, more general). Constructing invariants from partial information. Monovariants and termination proofs for iterative processes. The relationship between invariants and group actions on configurations.

Graphs, trees, connectivity at full depth. Euler tours, Hamiltonian paths and cycles, planarity and the Euler formula. Bipartite graphs and matchings. Hall’s marriage theorem with proof. König’s theorem, max-flow min-cut. Ramsey theory: Ramsey numbers, the general Ramsey theorem, multi-color Ramsey theory at conceptual level. Extremal graph theory: Turán’s theorem, the Erdős-Ko-Rado theorem.

Two-player games with perfect information at olympiad depth. Nim, Sprague-Grundy theorem with full proof. Computing Grundy values. Strategy stealing arguments and the games they resolve. Pairing strategies. Games on graphs and on game trees. Hackenbush and surreal numbers at conceptual level.

Existence proofs via random construction. Linearity of expectation as a proof tool. The deletion method and its applications. The Lovász local lemma with proof sketch and applications. Concentration inequalities: Markov, Chebyshev, Chernoff bounds at conceptual level. Random graphs and threshold phenomena.

Linear algebra over finite fields applied to counting and existence problems. The dimension argument. The polynomial method: encoding combinatorial structure as polynomial vanishing conditions. Combinatorial nullstellensatz. The Plünnecke-Ruzsa inequality in additive combinatorics. Recent results in algebraic combinatorics that have produced IMO problems.

Sumsets, sum-free sets, and the structure of additive sets. Cauchy-Davenport theorem with proof. The Erdős-Ginzburg-Ziv theorem and its generalizations. Schnirelmann density. Vinogradov’s three-primes theorem at conceptual level. Roth’s theorem on arithmetic progressions. The Szemerédi-Trotter theorem on point-line incidences.