USAJMO Prep for Younger Olympiad Mathletes
1-on-1 training with former IMO medalist coaches

Solving Cauchy functional equations f(x + y) = f(x) + f(y) and their multiplicative variants. Substitution patterns: setting x = 0, x = y, x = −y, x = 1 to extract structural information. Recognizing when continuity, monotonicity, or boundedness constraints force solutions to be linear or exponential. Functional equations on the rationals and on the integers — the discrete versus continuous distinction. Writing proofs that handle both the construction of a candidate solution and the uniqueness argument.

Vieta’s formulas at arbitrary degree. Newton’s identities relating power sums to elementary symmetric polynomials: p_k = e₁p_k−1 − e₂p_k−2 + … ± k · e_k. Multiplicity of roots and the derivative criterion. Polynomial interpolation via Lagrange’s formula. Schur’s inequality and other symmetric polynomial inequalities. Proving polynomial identities by evaluating at strategically chosen points.

The full inequality toolkit: AM-GM, GM-HM, QM-AM, Cauchy-Schwarz, rearrangement, Chebyshev, Jensen’s inequality for convex/concave functions. Weighted AM-GM. The tangent line trick and the SOS (sum of squares) method. Identifying equality cases — olympiad problems often hinge on when equality holds, not just on the inequality itself. Normalization and substitution techniques: WLOG assumptions, scaling, a + b + c = 1 normalizations.

Roots of unity and the roots-of-unity filter for extracting coefficients. Cyclotomic polynomials. Using complex numbers to prove real identities — the elegance of the complex approach. Representing geometric problems algebraically via complex coordinates: the plane as ℂ, rotations as multiplication, similar triangles as ratios. Proving classical geometry theorems via complex bash.

Linear recurrences and the characteristic equation. Repeated roots and complex roots, with their interpretations. Generating functions: ordinary (Σ a_nxⁿ) and exponential (Σ a_nxⁿ/n!). Solving counting problems by manipulating generating functions algebraically. The method of moments. Connecting generating functions to recurrences.

Fermat’s Little Theorem and Euler’s theorem with full proofs, not just statements. Wilson’s theorem and its converse. The Chinese Remainder Theorem with proof and applications. Primitive roots: existence, structure, applications. Quadratic residues and the Legendre symbol. Quadratic reciprocity (statement and use).

The p-adic valuation v_p(n). Basic properties: v_p(ab) = v_p(a) + v_p(b), v_p(a + b) ≥ min(v_p(a), v_p(b)) with equality when v_p(a) ≠ v_p(b). The Lifting the Exponent Lemma in full generality: for odd prime p with p | (a − b), p ∤ a, p ∤ b, we have v_p(aⁿ − bⁿ) = v_p(a − b) + v_p(n). Modified versions for p = 2 and for aⁿ + bⁿ. Recognizing LTE on USAJMO divisibility problems.

Linear and quadratic Diophantine equations. Pell’s equation x² − Dy² = 1 and its solution structure via fundamental solutions. The method of infinite descent: proving non-existence by constructing arbitrarily small solutions. Vieta jumping: a technique where one views a Diophantine equation as a quadratic in one variable and analyzes the relationship between roots. Vieta jumping appears on harder olympiad problems regularly.

Multiplicative arithmetic functions: τ(n), σ(n), φ(n), μ(n), and their convolutions. Möbius inversion. Computing sums Σ f(d) over divisors. Recognizing number-theoretic problems that decompose multiplicatively.

Proving the existence of integers with certain properties via construction. Proving non-existence via modular obstructions, size arguments, or descent. The interplay between additive and multiplicative structure. Bound arguments: showing a solution must be small or large, then checking finitely many cases.

Triangle centers and their interrelationships: centroid, circumcenter, incenter, orthocenter, the Euler line, the nine-point circle. Isogonal conjugates and isotomic conjugates. The Simson line. Configurations that recur on olympiad problems: the incircle and the contact triangle, the medial triangle, the orthic triangle, the tangential triangle.

Power of a Point in full generality. The radical axis of two circles. The radical center of three circles. Using power-of-a-point to prove concurrence, collinearity, and equal-length results. Coaxial circles and their properties.

Cyclic quadrilateral properties. Ptolemy’s theorem and Ptolemy’s inequality. Brahmagupta’s formula. Recognizing when four points are concyclic — a perennial olympiad sub-problem. Methods for proving concyclicity: equal angles, power-of-a-point converse, the converse of Ptolemy.

Rotations, reflections, translations, dilations, and their compositions. Spiral similarities and their fixed points. Using transformations to map a hard configuration to a known easy one. The technique of “transformation bashing” — reducing a problem to one where a clever transformation simplifies the structure.

Inversion in a circle: the basic definition and key properties. Circles and lines through the center invert to lines; circles not through the center invert to circles; angles are preserved. Using inversion to handle problems with multiple tangent circles, problems with concyclic points, or problems where a circle through a special point is featured. When inversion makes a problem trivial, and when it makes a problem harder.

Poles and polars. The cross-ratio of four collinear points. Harmonic conjugates and harmonic ranges. Projective transformations as a tool: mapping any four points to any four points. When projective methods solve a problem that synthetic methods can’t.

Coordinate geometry as a problem-solving tool. Cartesian coordinates, complex coordinates, barycentric coordinates — the choice depends on the problem. Barycentric coordinates for triangle-centric problems: representing points relative to the triangle’s vertices. Trigonometric identities applied to angle-chasing problems where pure synthetic methods are hard.

Proving that a locus is a particular curve. Constructing configurations satisfying multiple constraints. The interplay between algebraic and synthetic approaches. Proving when something is impossible to construct (e.g., trisecting an angle with compass and straightedge — a meta-result useful for problem context).

Bijections, double counting, and inclusion-exclusion at olympiad depth. Counting via complementary counting. Stars and bars in generalized form. Catalan numbers and their olympiad-relevant identities. Counting via algebraic manipulation: encoding a counting problem as a coefficient extraction problem.

The basic pigeonhole principle and its generalizations. The infinite pigeonhole. Pigeonhole on residues, on partitions, on configurations. Constructing the “pigeons” and “holes” — often the crucial step in an olympiad solution. Pigeonhole as a tool for proving existence by counting.

Considering the extremal element (largest, smallest, leftmost, with maximum/minimum value) to force structural properties. The well-ordering principle for the integers as the underlying reason extremal arguments work. Discrete extremal arguments for combinatorial games and configurations.

Invariants: quantities that remain unchanged under the moves allowed in a problem. Using invariants to prove impossibility — if the invariant of the starting position differs from the invariant of the target position, no sequence of moves connects them. Monovariants: quantities that strictly increase or decrease, used to prove termination of processes. Coloring arguments as a special case of invariants.

Graphs, paths, cycles, connectivity. Trees and their properties. Euler tours and Hamiltonian paths. Bipartite graphs and matchings. Hall’s marriage theorem. Ramsey theory at an introductory level: R(3, 3) = 6 and its proof. Using graph-theoretic structure to model combinatorial problems.

Game theory in the olympiad sense: two-player games with perfect information. Nim and the Sprague-Grundy theorem at an introductory level. Winning and losing positions, parity arguments. Strategy stealing. Pairing strategies for symmetric games.

Proving existence by showing that a randomly chosen object has positive probability of satisfying the desired property. Expected value arguments. Linearity of expectation as a proof tool, not just a computation. Examples from olympiad problems where the probabilistic method gives the cleanest proof.

Counting problems with geometric structure. Counting lattice points, lines, intersections. Convex hulls and their combinatorial properties. The Erdős-Szekeres theorem and related results. Helly’s theorem on convex sets in the plane.