$\Sigma$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

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

Double integrals by symmetry, Jacobians that collapse integrands, Lagrange multipliers, and Green's theorem in disguise

Direct proof, contradiction, and induction — as the graders want to see them

Quantities that survive every move, and symmetries that solve for you

Choose the largest, the smallest, the closest — and reason from the answer toward the start

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

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