The Syllabus, Honestly
Twelve problems, two three-hour sessions of six, each scored –. The median score is usually or ; a typically ranks in the top few hundred nationwide.
Almost all scores on a problem are , , , or — graders reward COMPLETE, rigorous solutions and give almost nothing for partial ideas. A-1 and B-1 are accessible to anyone who has finished this course; A-6 and B-6 are research-adjacent. The winning strategy is not to attempt everything: it is to find the two or three problems you can finish, write them so cleanly that a skeptical grader cannot object, and treat the rest as bonus.
ANALYSIS · LINEAR ALGEBRA · ALGEBRA & POLYNOMIALS · NUMBER THEORY · COMBINATORICS & PROBABILITY · GEOMETRY. Every exam samples all six.
Analysis: sequences and series, limits as Riemann sums or derivatives, the intermediate and mean value theorems, Taylor with remainder, and — the Putnam signature — definite integrals with no antiderivative. Linear algebra: structured determinants, eigenvalues and traces, rank, Cayley–Hamilton, matrices over finite fields. Algebra and polynomials: Vieta and Newton, roots of unity, interpolation, functional equations, the definitions of groups and rings. Number theory: orders, valuations, quadratic residues, Diophantine equations, divisor sums. Combinatorics and probability: bijections, generating functions, recurrences, inclusion–exclusion, expectation by indicators, random walks. Geometry: convexity, lattice points, complex numbers, vectors.
Topics 2–4 are the integrals. Topics 5–7 the rest of analysis. Topics 8–9 linear algebra. Topics 10–16 algebra, number theory, combinatorics, probability, abstract algebra, functional equations, geometry. Topic 17 is how to write a 10.
Every unit ends with a practice pool of hand-verified Putnam-level problems; the integral units share the non-elementary pool with the Calculus course. The Linear Algebra and Multivariable courses on this site are the prerequisites this course assumes.
Read a topic, then do its practice pool with a timer — forty minutes per problem, the exam pace. Write full solutions on paper even for numeric answers: the Putnam grades proofs, and the habit of justifying every step is the skill being trained here. Return to the Proofs sections until you can reproduce each argument from memory.