$\Sigma$Putnam

King's Rule, Queen's Rule and Glasser

Symmetry substitutions that make integrals evaluate themselves: reflect the interval (King), invert the variable (Queen), and Glasser's surprising theorem that ∫f(x − 1/x) = ∫f(x) over the whole line.

Frullani, Beta, Gamma and Weierstrass

The named integrals worth knowing by heart: Frullani's for differences, Beta and Gamma for powers and exponentials, Weierstrass's substitution for trig, series for logs, and when an improper integral converges at all.

Sequences and Series

Summing and testing series the Putnam way: telescoping, differentiating a geometric series, comparison and integral tests, alternating series, and the occasional sum that needs a clever regrouping.

Hard Sequences

Sequences the Putnam actually sets: closed forms hiding behind recurrences, growth rates found by squaring, and periodicity. Learn to guess the rate, then prove it with Stolz–Cesàro or a squeeze.

Limits, Continuity and the Big Theorems

The theorems that do the work on the Putnam: squeeze, the intermediate and mean value theorems, and the rigor behind them (uniform convergence, compactness), each shown on a problem it cracks.

Differential Equations on the Putnam

The ODE toolkit the exam assumes, equations that hide an ODE inside a functional equation, and qualitative arguments with comparison, phase lines and blow-up, where you prove what a solution does without solving it.

Inequalities and Convexity

The Putnam inequality toolkit: convexity and Jensen, AM–GM and Cauchy–Schwarz, Young and Hölder, and calculus as the final resort. Plus the classic: which is bigger, e^π or π^e?

Determinants the Putnam Way

Evaluate determinants without expanding: row-reduce, spot the Vandermonde or circulant structure, use the matrix determinant lemma, and read off parity. Patterns the Putnam rewards over brute force.

Spectra, Cayley–Hamilton and Finite Fields

Reason with eigenvalues instead of entries: Cayley–Hamilton to compute powers, trace and determinant as sum and product, nilpotent and idempotent matrices, and matrices over finite fields.

Polynomials: Roots You Never Find

Answer questions about roots without solving for them: Vieta, double roots through the derivative, the roots-of-unity filter, and the divisibility rule a − b | P(a) − P(b) that kills impossible integer polynomials.

Proof-Level Number Theory

Fermat and Wilson, orders and primitive roots, and lifting the exponent: the number theory a Putnam proof is built from, with every tool used on a real problem.

Counting With Structure

Identities you can see in Pascal's triangle, generating functions that count coins and partitions, Catalan bijections, and double counting: the combinatorics a Putnam solution leans on.

Expectation Without Distributions

Linearity of expectation and indicators, geometric probability, and first-step analysis: compute expected values without ever finding the full distribution.

Abstract Algebra the Putnam Assumes

Groups, subgroups and Lagrange's theorem, rings and fields mod n, and the classic short proofs: just the definitions the Putnam takes for granted, each put to work.

Solving for Functions

Cauchy's equation and what regularity buys, substitutions that turn a functional equation into a linear system, linear recurrences via characteristic roots, and the logarithm equation.

Multivariable Moves That Recur

Polar coordinates and the Gaussian integral, Jacobians for clever substitutions, Lagrange multipliers on a constraint, and swapping the order of integration: the multivariable tricks that keep coming back on the Putnam.

Proof Techniques at Putnam Level

Direct proof, contradiction and contrapositive; induction made easier by proving more; existence by construction or by trapping; and the minimal counterexample.

Invariants, Monovariants and Symmetry

Find the quantity that never changes, the one that only goes down, or the symmetry that pairs things up. Colourings, parity, Nim and a famous integral.

Extremal Principle and Working Backwards

Look at the biggest, smallest or longest thing; descend forever and contradict; jump between roots with Vieta; and solve games from the end.

Strategy and Writing a 10

How the exam works, why small cases come first (and when patterns lie), and what separates a 10 from a 2 on the page.