Putnam

The complete map of the exam, and where each piece lives in this course

Differentiation under the integral sign — the tool that makes impossible integrals routine

Substitutions that map the interval onto itself, then add

The named results that finish what symmetry and parameters start

The second shelf of named integrals, each with its proof

Convergence, telescoping, power series, and Stolz–Cesàro

Squaring tricks, reciprocal tricks, Lucas closed forms, Stolz rates, and maps of finite order

IVT, MVT, Taylor with remainder, and limits in disguise

Separable and linear equations, uniqueness by integrating factor, blow-up, and Grönwall

AM–GM, Cauchy–Schwarz, Jensen, rearrangement, and integral inequalities

Rank-one updates, Vandermonde, circulants, tridiagonals

Eigenvalues do the arithmetic; counting matrices over F_p

Vieta, Newton, roots of unity, interpolation, integer polynomials

Orders, valuations, quadratic residues, Diophantine equations, divisor sums

Bijections, recurrences, generating functions, inclusion–exclusion, extremal arguments

Indicators, linearity, geometric probability, random walks, coupon collecting

Groups, orders, Lagrange, cyclic groups, rings and fields at the Putnam level

Substitution, symmetry, Cauchy’s equation, periodicity, fixed points

Convexity, lattice points, complex numbers, vectors

Choosing problems, managing time, and writing proofs graders cannot dent

The Putnam reference sheet: analysis, algebra, and the named integrals in one place