Putnam Exam Prep for College Mathletes
1-on-1 coaching from former IMC & IMO medalists

Polynomial identities, roots, and their relationships at undergraduate depth. Vieta’s formulas in their full generality. Newton’s identities relating power sums to elementary symmetric polynomials. Polynomial interpolation: Lagrange, Newton’s divided differences, and the relationship between interpolation and finite differences. Resultants of polynomials and their use in eliminating variables. The fundamental theorem of algebra and its consequences for olympiad-style problems.

Vector spaces, linear maps, matrices. Determinants and their properties: multilinearity, expansion by minors, Cramer’s rule. Eigenvalues, eigenvectors, characteristic polynomials. The Cayley-Hamilton theorem. Diagonalization and the Jordan canonical form at conceptual level. Computing matrix powers via diagonalization. Trace and its relationship to eigenvalues. The Spectral Theorem for symmetric matrices. Linear algebra appears on roughly one to two Putnam problems each year, often in unusual contexts.

The standard inequalities (AM-GM, Cauchy-Schwarz, Jensen, Hölder, Minkowski) at undergraduate depth. The Cauchy-Schwarz inequality in inner product spaces. Convex functions and Jensen’s inequality. Integral inequalities and their relationship to discrete inequalities. The technique of recognizing when a Putnam inequality problem reduces to a known classical result versus when it requires a novel argument. Convexity arguments at the level of real-valued functions on intervals.

Complex numbers, analytic functions, the Cauchy-Riemann equations at conceptual level. Power series and their radii of convergence. Roots of unity in their full generality. Cyclotomic polynomials and their properties. Using complex analysis as a tool for evaluating real sums and integrals — a technique that occasionally appears on Putnam. The relationship between complex integration and real analysis at problem-solving depth.

Rigorous treatment of limits at undergraduate depth. Continuity and its consequences. The intermediate value theorem, the extreme value theorem, and their applications to existence proofs. Differentiation and its rules. Mean value theorems (Rolle, Lagrange, Cauchy). Taylor’s theorem with remainder and its applications. L’Hôpital’s rule and its limitations.

The Riemann integral and its properties. The fundamental theorem of calculus. Techniques of integration: substitution, integration by parts, partial fractions, trigonometric substitutions. Improper integrals and convergence tests. Computing definite integrals via clever substitutions, symmetry arguments, or differentiation under the integral sign (the Feynman technique). Recognizing when a Putnam integral problem has an elegant solution via a non-obvious method.

Convergence and divergence of sequences and series. Tests for convergence: ratio, root, integral, comparison, alternating series, Abel’s and Dirichlet’s tests. Power series and Taylor series. Computing limits of sequences defined recursively. Asymptotic estimates for sequences and series. The relationship between summation and integration via comparison.

Multivariable calculus at the level needed for Putnam. Partial derivatives, gradients, the chain rule. Multiple integrals and changes of variables (the Jacobian). Line integrals and surface integrals at introductory level. Optimization with and without constraints (Lagrange multipliers). Differential geometry of curves and surfaces at conceptual level for problems where geometric intuition matters.

First-order ODEs and their solution methods. Second-order linear ODEs with constant coefficients. The relationship between ODEs and recurrence relations. Phase plane analysis at conceptual level. ODEs occasionally appear on Putnam, particularly in modeling problems where setting up the equation is the key insight.

Fermat’s Little Theorem, Euler’s theorem, Wilson’s theorem at full depth. The Chinese Remainder Theorem. Primitive roots, quadratic residues, quadratic reciprocity. The structure of (ℤ/pℤ)*. These foundational theorems appear regularly on Putnam in subtle disguises.

Algebraic integers in quadratic fields ℤ[√d] and ℤ[ω]. Norms and traces in number fields. Unique factorization and its failure. Euclidean domains, principal ideal domains, unique factorization domains. The Gaussian integers and their applications to problems involving sums of squares.

Linear and quadratic Diophantine equations. Pell’s equation and continued fractions. Vieta jumping. The method of infinite descent. Recognizing Diophantine structure in problems that don’t announce themselves as number-theoretic.

Bijective arguments, double counting, generating functions. Inclusion-exclusion with applications. Catalan numbers and their many incarnations. Recurrence relations and their solutions via generating functions. Counting via algebraic methods.

Graphs, trees, connectivity. Euler tours, Hamiltonian paths. Ramsey theory at introductory level. Matchings and Hall’s marriage theorem. Algebraic graph theory: the adjacency matrix and the spectrum of a graph at conceptual level. Putnam includes graph theory roughly every other year.

Discrete probability spaces and their structure. Expected value and linearity of expectation as proof techniques. Conditional probability and Bayes’ theorem. Common probability distributions and their moments. The probabilistic method in combinatorics. Putnam probability problems often have elegant solutions via linearity of expectation.

Triangle centers and their relationships. The Euler line, the nine-point circle. Power of a point and the radical axis. Cyclic quadrilaterals and Ptolemy’s theorem. Inversion and its applications. Projective geometry at conceptual level. While pure geometry appears less often on Putnam than on IMO, when it does appear it’s usually at this level of depth.

Cartesian, polar, and complex coordinates. Conic sections and their properties. Quadric surfaces in 3D. Geometric transformations (isometries, similarities, affine transformations). Differential geometry of curves: arc length, curvature, the Frenet-Serret formulas at introductory level.

Counting problems with geometric structure. Lattice points, convex hulls. The Erdős-Szekeres theorem. Helly’s theorem. The Sylvester-Gallai theorem. Geometric Ramsey theory at introductory level.