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Every rule and concept in Putnam, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (131)

Absolute versus conditional, and the principal value: $\int |f|$ convergent implies $\int f$ convergent.; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Improper Integrals: When They Converge
Bijections before formulas: Pair objects with objects.; Combinatorics & Generating Functions › Counting With Structure
Cauchy's equation and its relatives: $f(x + y) = f(x) + f(y)$ has only $f(x) = cx$ on $\mathbb Q$ (and on $\mathbb R$ under any regularity: continuous, monotone, bounded on an interval).; Functional Equations & Recurrences › Solving for Functions
Cayley–Hamilton and minimal polynomials: Every matrix satisfies its own characteristic polynomial: $p_A(A) = 0$ .; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Spectral Reasoning
Change variables to match the integrand: With $u = x + y$ , $v = x - y$ (Jacobian $\tfrac12$ ), or polar coordinates, or $u = xy$ , $v = y/x$ : choose the substitution that makes the integrand a function of one new variable.; Multivariable Calculus on the Putnam › The Moves That Recur
Choose the extreme, then read off an inequality: Take the object that is largest or smallest in some measure — a minimum label, a closest pair, a point nearest a line.; Extremal Principle and Working Backwards › The Extremal Principle, Worked
Classify states from the end: When a process or game ends in known states, classify every state by working backwards from them: in a game, a position is a loss for the player to move (a P-position) iff every move leads to a win for the opponent (an N-position), and a win iff SOME move l…; Extremal Principle and Working Backwards › Working Backwards, Worked
Closed forms from $\alpha + \alpha^{-1}$: If $a_0 = \alpha + \alpha^{-1}$ and $a_{n+1} = a_n^2 - 2$ , then $a_n = \alpha^{2^n} + \alpha^{-2^n}$ .; Hard Sequences: Closed Forms, Rates & Periodicity › Sequences the Putnam Actually Sets
Combinatorics and probability: Bijections; PIE; derangements; Catalan and the reflection principle; generating functions and $x\tfrac{d}{dx}$ ; Erdős–Szekeres; Mantel/Turán, Hall, Euler; extremal and pigeonhole; games (P/N, Nim, strategy stealing).; Formulas & Methods › Combinatorics, Probability & Geometry
Compactness, and why it is the hypothesis that saves proofs: A closed bounded subset of $\mathbb{R}^n$ is compact; a continuous function on a compact set attains its maximum and minimum, and is uniformly continuous there.; Limits, Continuity & the Big Theorems › Rigor: $\varepsilon$ - $\delta$ , Uniform Convergence and Compactness
Comparison and limit comparison: If $0 \le f \le g$ and $\int g$ converges, so does $\int f$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Improper Integrals: When They Converge
Compute, conjecture, prove: Compute the first few cases EXACTLY (fractions, not decimals).; Strategy & Writing a 10 › Small Cases, Then Generalize, Worked
Conditioning and first-step analysis: For expected times and absorption probabilities, condition on the first step: $E_i = 1 + \sum_jp_{ij}E_j$ — a small linear system.; Probability & Expectation › Expectation Without Distributions
Constant-coefficient linear equations: $y'' + py' + qy = 0$ has solutions $e^{rx}$ for the roots $r$ of $r^2 + pr + q = 0$ (with $xe^{rx}$ for a double root, and $e^{ax}\cos bx$ , $e^{ax}\sin bx$ for $r = a \pm bi$ ).; Differential Equations on the Putnam › The ODE Toolkit the Exam Assumes
Contradiction with a minimal counterexample: Assume the claim fails; take the SMALLEST failure (smallest $n$ , shortest sequence, least degree).; Proof Techniques for the Putnam › The Three Forms, at Putnam Level
Convexity and its consequences: A function with $f'' \ge 0$ lies above its tangent lines and below its chords.; Inequalities & Convexity › The Inequality Toolkit
Coordinates adapted to the integrand: If the integrand depends on $x$ and $y$ only through some combinations $u(x, y)$ , $v(x, y)$ , integrate in $u, v$ : $\iint_Rf\,dx\,dy = \iint_{R'}f\,|J|\,du\,dv$ with $J = \dfrac{\partial(x, y)}{\partial(u, v)}$ .; Multivariable Calculus on the Putnam › Double Integrals by Substitution, Worked
Differentiate the condition, then read off initial values: A condition involving $\int_0^x$ or a shifted argument such as $f(1 - x)$ often becomes an ordinary differential equation after one or two differentiations.; Differential Equations on the Putnam › Equations That Hide an ODE
Direct proofs that name their theorems: A direct proof at this level is a chain of cited results: "by the mean value theorem there is $c \in (a, b)$ with…", "since the series converges absolutely we may rearrange…".; Proof Techniques for the Putnam › The Three Forms, at Putnam Level
Dirichlet variants by parts: $\int_0^\infty\frac{\sin^2x}{x^2}dx = \int_0^\infty\frac{1 - \cos x}{x^2}dx = \frac\pi2$ — one integration by parts each, landing on $\int_0^\infty\frac{\sin x}{x}dx$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › More Named Results
Divisor sums and multiplicative functions: $\tau$ , $\sigma$ , $\varphi$ are multiplicative: compute on prime powers and multiply.; Number Theory › Proof-Level Number Theory
Eigenvalues carry every polynomial fact: If $A$ has eigenvalues $\lambda_i$ then $p(A)$ has eigenvalues $p(\lambda_i)$ : $\det(A^2 + I) = \prod(\lambda_i^2 + 1)$ , $\mathrm{tr}(A^k) = \sum\lambda_i^k$ , $\det e^A = e^{\mathrm{tr}A}$ .; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Spectral Reasoning
Fresnel, the reflection formula, and Wallis: $\int_0^\infty\sin(x^2)dx = \int_0^\infty\cos(x^2)dx = \frac12\sqrt{\frac\pi2}$ ; $\Gamma(s)\Gamma(1 - s) = \frac{\pi}{\sin\pi s}$ ; $\prod_{n\ge1}\frac{4n^2}{4n^2 - 1} = \frac\pi2$ ; $\int_0^1\frac{\ln x}{1 - x}dx = -\frac{\pi^2}{6}$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › More Named Results
Frullani's theorem: $\displaystyle\int_0^\infty\frac{f(ax) - f(bx)}{x}\,dx = \big(f(0) - f(\infty)\big)\ln\frac ba$ for continuous $f$ with finite limits at $0$ and $\infty$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › The Named Integrals
Gamma and Beta: $\Gamma(s) = \displaystyle\int_0^\infty x^{s-1}e^{-x}dx$ , with $\Gamma(n+1) = n!$ and $\Gamma(\tfrac12) = \sqrt\pi$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › The Named Integrals
Geometric probability and random walks: Probability $=$ favorable measure $\div$ total measure; expectations become integrals.; Probability & Expectation › Expectation Without Distributions
Geometry: Pick's theorem; $\gcd + 1$ lattice points on a segment; Minkowski; Gauss circle count $\pi r^2 + O(r)$ ; complex numbers (rotation, collinearity via real ratios, equilateral criterion $a + \omega b + \omega^2c = 0$ ); point-to-line/plane distances; convexity …; Formulas & Methods › Combinatorics, Probability & Geometry
Glasser's master theorem: $\displaystyle\int_{-\infty}^\infty f\!\left(x - \sum_{i}\frac{a_i}{x - b_i}\right)dx = \int_{-\infty}^\infty f(x)\,dx$ for $a_i > 0$ .; Integrals II: King's Rule, Queen's Rule, and Glasser › Symmetry Substitutions
Green, Stokes and the divergence theorem as identities: A line integral around a closed curve equals a double integral of a curl; the flux of $\mathbf r/|\mathbf r|^3$ through any surface enclosing the origin is $4\pi$ ; the area of a region is $\tfrac12\oint(x\,dy - y\,dx)$ .; Multivariable Calculus on the Putnam › The Moves That Recur
Groups and orders: A group is a set with an associative operation, an identity, and inverses.; Abstract Algebra Basics › The Definitions the Putnam Assumes
Identity plus rank one: $\det(I + \mathbf u\mathbf v^{\mathsf T}) = 1 + \mathbf v^{\mathsf T}\mathbf u$ (the matrix determinant lemma).; Linear Algebra I: Determinants & Structured Matrices › Determinants the Putnam Way
Inclusion–exclusion and extremal arguments: $|A_1\cup\cdots\cup A_n| = \sum|A_i| - \sum|A_i\cap A_j| + \cdots$ .; Combinatorics & Generating Functions › Counting With Structure
Induction that is actually strong: When $P(n + 1)$ depends on several earlier cases, assume $P(k)$ for ALL $k \le n$ .; Proof Techniques for the Putnam › The Three Forms, at Putnam Level
Inequalities and convexity: AM–GM, Cauchy–Schwarz (sums and integrals), Jensen, rearrangement, power means; Hermite–Hadamard $f(\tfrac{a+b}{2}) \le \tfrac{1}{b-a}\int_a^bf \le \tfrac{f(a) + f(b)}{2}$ for convex $f$ ; tangent-line trick; MVT-derived bounds ( $e^x \ge 1 + x$ , $ x ≤ x…; Formulas & Methods › Analysis
Integer polynomials and interpolation: For $P \in \mathbb Z[x]$ : $a - b \mid P(a) - P(b)$ .; Polynomials › Working With Roots You Never Find
IVT, MVT, Rolle, and Taylor with remainder: Continuous $f$ on $[a,b]$ takes every value between $f(a)$ and $f(b)$ .; Limits, Continuity & the Big Theorems › The Theorems That Do the Work
Jump to the other root: For an equation that is quadratic in each variable, fix all but one variable, say $x$ ; if $x = a$ is a root of the quadratic $x^2 - px + q = 0$ , so is $a' = p - a = \tfrac qa$ .; Extremal Principle and Working Backwards › Vieta Jumping, Worked
King's rule: $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ — reflect the interval onto itself.; Integrals II: King's Rule, Queen's Rule, and Glasser › Symmetry Substitutions
Know the solution without solving: Phase line: for $y' = g(y)$ , a solution never crosses an equilibrium ( $g(c) = 0$ ), and between equilibria it is monotone.; Differential Equations on the Putnam › Comparison, Phase Lines and Blow-Up
Lagrange multipliers, then check the boundary: To extremize $f$ on $g = c$ : solve $\nabla f = \lambda\nabla g$ together with $g = c$ .; Multivariable Calculus on the Putnam › The Moves That Recur
Limits that are derivatives, integrals, or exponentials in disguise: $n(2^{1/n} - 1) \to \ln 2$ (a derivative); $\tfrac1n\sum f(\tfrac kn) \to \int_0^1 f$ (an integral); $(1 + \tfrac an)^n \to e^a$ (an exponential); $\tfrac{(n!)^{1/n}}{n} \to \tfrac1e$ (a log of a Riemann sum).; Limits, Continuity & the Big Theorems › The Theorems That Do the Work
Limits, series, integrals: Limits as derivatives ( $n(a^{1/n} - 1) \to \ln a$ ), as Riemann sums ( $\tfrac1n\sum f(\tfrac kn) \to \int_0^1f$ ), as exponentials ( $(1 + \tfrac an)^n \to e^a$ ); Stolz–Cesàro; Taylor with Lagrange remainder; the three series ( $\sum x^n$ , $\sum\tfrac{x^n}{n}$ ,…; Formulas & Methods › Analysis
Linear algebra: Matrix determinant lemma $\det(I + \mathbf u\mathbf v^{\mathsf T}) = 1 + \mathbf v^{\mathsf T}\mathbf u$ and $\det(aI + bJ) = a^{n-1}(a + nb)$ ; Vandermonde $\prod_{i<j}(x_j - x_i)$ ; circulant eigenvalues by DFT; tridiagonal Toeplitz eigenvalues $a + 2b\…; Formulas & Methods › Algebra
Linearity and indicators: $E[\sum X_i] = \sum E[X_i]$ with NO independence assumption.; Probability & Expectation › Expectation Without Distributions
Log-trig integrals and the tangent bridge: $\int_0^{\pi/2}\ln\cos x\,dx = -\frac\pi2\ln 2$ , hence $\int_0^\infty\frac{\ln(1 + x^2)}{1 + x^2}dx = \pi\ln 2$ (substitute $x = \tan\theta$ ) and $\int_0^{\pi/2}\ln\tan x\,dx = 0$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › More Named Results
Maps of finite order and periodic sequences: $x \mapsto \tfrac{1}{1-x}$ has order $3$ ; $x \mapsto \tfrac{1+x}{1-x}$ has order $4$ ; $x \mapsto \tfrac{x+1}{x-1}$ is an involution.; Hard Sequences: Closed Forms, Rates & Periodicity › Sequences the Putnam Actually Sets
Matrices over finite fields: count: $|GL_n(\mathbb F_q)| = (q^n - 1)(q^n - q)\cdots(q^n - q^{n-1})$ — pick columns one at a time, each outside the span of the previous.; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Spectral Reasoning
Monovariants: A quantity that always DEcreases (or increases) and is bounded proves that a process terminates, and — if it is an integer — how many steps at most.; Invariance and Symmetry › Finding the Invariant
Orders, primitive roots, and CRT: The order of $a$ mod $n$ divides every $k$ with $a^k \equiv 1$ , hence divides $\varphi(n)$ .; Number Theory › Proof-Level Number Theory
Pair, cancel, mirror: A symmetry of the problem — swapping players, swapping heads and tails, reflecting through a point, $x \mapsto a + b - x$ — maps outcomes to outcomes.; Invariance and Symmetry › Symmetry Arguments, Worked
Parity, sums mod $k$ , and colorings: For a process (a game, an operation repeated on a board or a set), compute a quantity after each move: the parity of a count, a sum modulo $k$ , a weighted sum, a determinant, a product of signs.; Invariance and Symmetry › Finding the Invariant
Periodicity, involutions, and iteration: Compose the function with itself; find the order.; Functional Equations & Recurrences › Solving for Functions
Pick the extreme object: Among finitely many objects, choose the one maximizing or minimizing something — the largest number, the point farthest from a line, the longest chain, the polynomial of least degree.; Extremal Principle and Working Backwards › The Extremal Principle
Pointwise versus uniform convergence: Pointwise: for each $x$ , $f_n(x) \to f(x)$ .; Limits, Continuity & the Big Theorems › Rigor: $\varepsilon$ - $\delta$ , Uniform Convergence and Compactness
Polynomials, number theory, abstract algebra: Vieta and Newton; roots of unity filter; auxiliary polynomial with known roots for values at integers; Chebyshev; Eisenstein; $a - b \mid P(a) - P(b)$ .; Formulas & Methods › Algebra
Proof: $\frac{(n!)^{1/n}}{n} \to \frac1e$: A limit that looks like it needs Stirling and does not.; Limits, Continuity & the Big Theorems › Proofs & Why It Matters
Proof: $\Gamma(\tfrac12) = \sqrt\pi$ and the Gaussian integral: Two famous constants, one computation.; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Proofs & Why It Matters
Proof: $\int_0^\infty e^{-tx}\sin x\,dx = \frac{1}{1+t^2}$: The Laplace transform that powers the Dirichlet integral.; Integrals I: Feynman's Trick › Proofs & Why It Matters
Proof: $\int_0^\infty\frac{\sin^2x}{x^2}dx = \frac\pi2$: Parts onto Dirichlet.; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Proofs & Why It Matters
Proof: $\operatorname{tr}A^k = 0$ for $k = 1, \ldots, n$ forces $A$ nilpotent: If an $n\times n$ complex matrix has $\operatorname{tr}A^k = 0$ for all $1 \le k \le n$ , then every eigenvalue is $0$ .; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Proofs & Why It Matters
Proof: $\sum_{d\mid n}\varphi(d) = n$: A one-line counting proof of a multiplicative identity.; Number Theory › Proofs & Why It Matters
Proof: $\zeta(2) = \frac{\pi^2}{6}$ via a double integral: The Basel problem, by a route the Putnam would accept.; Sequences & Series › Proofs & Why It Matters
Proof: a polynomial over a field has at most $\deg$ roots: The property that separates fields from rings like $\mathbb Z/8$ .; Abstract Algebra Basics › Proofs & Why It Matters
Proof: an increasing involution is the identity: If $f$ is strictly increasing and $f(f(x)) = x$ for all $x$ , then $f(x) = x$ .; Functional Equations & Recurrences › Proofs & Why It Matters
Proof: blow-up for $y' \ge y^2$: If $y$ is differentiable with $y' \ge y^2$ and $y(0) = c > 0$ , then $y$ cannot exist on $[0, 1/c]$ .; Differential Equations on the Putnam › Proofs & Why It Matters
Proof: Cauchy–Schwarz by a quadratic: For real vectors $\mathbf a, \mathbf b$ : $(\mathbf a\cdot\mathbf b)^2 \le |\mathbf a|^2|\mathbf b|^2$ .; Inequalities & Convexity › Proofs & Why It Matters
Proof: Cauchy's equation on $\mathbb Q$: If $f(x + y) = f(x) + f(y)$ for all rationals, then $f(x) = cx$ with $c = f(1)$ .; Functional Equations & Recurrences › Proofs & Why It Matters
Proof: Cayley–Hamilton via the adjugate: Every square matrix satisfies its own characteristic polynomial: $p_A(A) = 0$ .; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Proofs & Why It Matters
Proof: Erdős–Szekeres: Any sequence of $n^2 + 1$ distinct reals has a monotone subsequence of length $n + 1$ .; Combinatorics & Generating Functions › Proofs & Why It Matters
Proof: Frullani's theorem: Assume $f$ continuous with $f(0)$ , $f(\infty)$ finite and $0 < a < b$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Proofs & Why It Matters
Proof: Glasser's theorem in the simplest case: Show $\displaystyle\int_{-\infty}^\infty f\!\left(x - \tfrac1x\right)dx = \int_{-\infty}^\infty f(u)\,du$ .; Integrals II: King's Rule, Queen's Rule, and Glasser › Proofs & Why It Matters
Proof: Grönwall's inequality: If $u'(x) \le \beta u(x)$ on $[0, T]$ with $u(0) = u_0$ , then $u(x) \le u_0e^{\beta x}$ .; Differential Equations on the Putnam › Proofs & Why It Matters
Proof: Jensen's inequality: For convex $f$ and weights $w_i \ge 0$ summing to $1$ : $f(\sum w_ix_i) \le \sum w_if(x_i)$ .; Inequalities & Convexity › Proofs & Why It Matters
Proof: Lagrange's theorem: For a subgroup $H$ of a finite group $G$ , $|H|$ divides $|G|$ .; Abstract Algebra Basics › Proofs & Why It Matters
Proof: lifting the exponent (odd prime case): For odd prime $p$ with $p \mid a - b$ and $p \nmid ab$ : $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ .; Number Theory › Proofs & Why It Matters
Proof: linearity of expectation: $E[X + Y] = E[X] + E[Y]$ for ANY random variables with finite expectations.; Probability & Expectation › Proofs & Why It Matters
Proof: Newton's identity for $p_2$ and the general pattern: $p_2 = e_1^2 - 2e_2$ , and in general $p_k - e_1p_{k-1} + e_2p_{k-2} - \cdots + (-1)^kke_k = 0$ .; Polynomials › Proofs & Why It Matters
Proof: Stolz–Cesàro: If $b_n$ is strictly increasing to $\infty$ and $\tfrac{a_{n+1} - a_n}{b_{n+1} - b_n} \to L$ , then $\tfrac{a_n}{b_n} \to L$ .; Sequences & Series › Proofs & Why It Matters
Proof: Stolz–Cesàro: If $b_n$ is strictly increasing with $b_n \to \infty$ and $\tfrac{a_{n+1} - a_n}{b_{n+1} - b_n} \to L$ , then $\tfrac{a_n}{b_n} \to L$ .; Hard Sequences: Closed Forms, Rates & Periodicity › Proofs & Why It Matters
Proof: the Catalan reflection: Lattice paths from $(0,0)$ to $(n,n)$ never rising above $y = x$ number $\tfrac{1}{n+1}\binom{2n}{n}$ .; Combinatorics & Generating Functions › Proofs & Why It Matters
Proof: the coupon collector expectation: Collecting all $n$ equally likely coupons takes $nH_n$ draws on average.; Probability & Expectation › Proofs & Why It Matters
Proof: the Leibniz rule: If $f$ and $\partial_af$ are continuous on $[c,d]\times[a_0, a_1]$ , then $\dfrac{d}{da}\int_c^d f(x,a)\,dx = \int_c^d\partial_af(x,a)\,dx$ .; Integrals I: Feynman's Trick › Proofs & Why It Matters
Proof: the log-sine integral: Show $\displaystyle\int_0^{\pi/2}\ln(\sin x)\,dx = -\frac\pi2\ln 2$ using only King and Queen.; Integrals II: King's Rule, Queen's Rule, and Glasser › Proofs & Why It Matters
Proof: the matrix determinant lemma: $\det(I + \mathbf u\mathbf v^{\mathsf T}) = 1 + \mathbf v^{\mathsf T}\mathbf u$ , and more generally $\det(A + \mathbf u\mathbf v^{\mathsf T}) = \det A\,(1 + \mathbf v^{\mathsf T}A^{-1}\mathbf u)$ for invertible $A$ .; Linear Algebra I: Determinants & Structured Matrices › Proofs & Why It Matters
Proof: the mean value theorem from Rolle: Rolle: if $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a) = f(b)$ , then $f'(c) = 0$ for some $c$ .; Limits, Continuity & the Big Theorems › Proofs & Why It Matters
Proof: the roots-of-unity filter: For $P(x) = \sum a_kx^k$ and $\omega = e^{2\pi i/n}$ : $\sum_{n\mid k}a_k = \tfrac1n\sum_{j=0}^{n-1}P(\omega^j)$ .; Polynomials › Proofs & Why It Matters
Proof: the spectral theorem for real symmetric matrices: A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors.; Linear Algebra II: Spectra, Cayley–Hamilton & Finite Fields › Proofs & Why It Matters
Proof: the tridiagonal recurrence and its eigenvalues: For $T_n$ with $a$ on the diagonal and $b, c$ on the off-diagonals, $\det T_n = a\det T_{n-1} - bc\det T_{n-2}$ ; the eigenvalues of the symmetric case ( $b = c$ ) are $a + 2b\cos\frac{k\pi}{n+1}$ , $k = 1, \ldots, n$ .; Linear Algebra I: Determinants & Structured Matrices › Proofs & Why It Matters
Proof: the Vandermonde determinant: $\det\big[x_i^{\,j-1}\big]_{i,j=1}^n = \prod_{1\le i<j\le n}(x_j - x_i)$ .; Linear Algebra I: Determinants & Structured Matrices › Proofs & Why It Matters
Prove more, so that the step closes: If the step $P(n) \Rightarrow P(n+1)$ will not go through, look for a STRONGER statement $Q(n)$ — one that implies $P(n)$ — whose own step does go through.; Proof Techniques for the Putnam › Strengthening the Induction Hypothesis
Queen's rule and Jack's rule: $\displaystyle\int_0^{2a} f = 2\int_0^a f$ if $f(2a - x) = f(x)$ , and $= 0$ if $f(2a - x) = -f(x)$ .; Integrals II: King's Rule, Queen's Rule, and Glasser › Symmetry Substitutions
Recurrences and generating functions: Set up $a_n$ by the first choice; solve by characteristic equation or by the generating function $A(x) = \sum a_nx^n$ , which turns the recurrence into algebra.; Combinatorics & Generating Functions › Counting With Structure
Recurring Putnam groups: Symmetric groups $S_n$ (permutations, cycle structure, parity), matrix groups $GL_n(\mathbb F_q)$ , units mod $n$ , dihedral groups, and "the set of all maps satisfying …" that turns out to be a group.; Abstract Algebra Basics › The Definitions the Putnam Assumes
Rings, fields, and polynomials over them: A field has division by nonzero elements; $\mathbb F_p = \mathbb Z/p$ and its extensions $\mathbb F_{p^k}$ are the finite ones.; Abstract Algebra Basics › The Definitions the Putnam Assumes
Roots of unity and the filter: $\sum_{k=0}^{n-1}\omega^{jk} = n$ if $n \mid j$ , else $0$ .; Polynomials › Working With Roots You Never Find
Sentences graders reward: "We proceed by strong induction on $n$ ." "Suppose for contradiction that no such $x$ exists; let $n_0$ be the least…" "By the intermediate value theorem applied to the continuous function $g$ on $[0, 1]$ …" "The case $n = 1$ holds since…"; Proof Techniques for the Putnam › The Three Forms, at Putnam Level
Separable and first-order linear equations: Separable: $y' = f(x)g(y) \Rightarrow \int\tfrac{dy}{g(y)} = \int f(x)dx$ .; Differential Equations on the Putnam › The ODE Toolkit the Exam Assumes
Small cases, then generalize: Compute $n = 1, 2, 3$ .; Strategy & Writing a 10 › How to Take the Exam
Stolz–Cesàro and Riemann sums: If $b_n \uparrow \infty$ and $\dfrac{a_{n+1} - a_n}{b_{n+1} - b_n} \to L$ , then $\dfrac{a_n}{b_n} \to L$ .; Sequences & Series › Summing and Testing
Symmetrize before integrating: Over a region symmetric under $x \leftrightarrow y$ , $\iint f(x, y) = \iint f(y, x)$ ; average the two.; Multivariable Calculus on the Putnam › The Moves That Recur
Symmetry that solves the problem: If the problem is invariant under a transformation (swapping variables, $x \to 1 - x$ , rotation of a configuration), the answer or the extremal configuration is often invariant too — or the transformation pairs up terms that cancel.; Invariance and Symmetry › Finding the Invariant
Tangency, then compare the candidates: At an extremum of $f$ on the curve (or surface) $g = c$ , at a point where $\nabla g \ne 0$ : $\nabla f = \lambda\nabla g$ for some $\lambda$ .; Multivariable Calculus on the Putnam › Constrained Extrema, Worked
Telescoping and partial fractions: Write the term as $g(n) - g(n+1)$ (or a lagged version) and watch everything cancel except the ends.; Sequences & Series › Summing and Testing
Termination by a monovariant: If every move strictly increases (or decreases) a quantity $M$ , and $M$ can take only finitely many values — or values in a set with no infinite decreasing chain, such as the nonnegative integers — then the process stops.; Invariance and Symmetry › Monovariants, Worked
The core four: AM–GM: $\tfrac{\sum a_i}{n} \ge \sqrt[n]{\prod a_i}$ .; Inequalities & Convexity › The Inequality Toolkit
The first move in each pool: Linear algebra: row operations before any formula.; Everything the Putnam Tests › One Problem From Each Pool
The invariant method: Find a quantity $I$ with $I(\text{after}) = I(\text{before})$ for EVERY allowed move.; Invariance and Symmetry › Invariants, Worked
The map of this course: Topics 2–4 are the integrals.; Everything the Putnam Tests › The Syllabus, Honestly
The method in one line: Embed the integral in a family $I(a) = \int f(x, a)\,dx$ , differentiate in the parameter, integrate back, and fix the constant from a value of $a$ where $I(a)$ is known.; Integrals I: Feynman's Trick › Differentiating Under the Integral
The named integrals: Feynman: $I'(a) = \int\partial_af$ ; Dirichlet $\int_0^\infty\tfrac{\sin x}{x} = \tfrac\pi2$ and $\int_0^\infty\tfrac{\sin^2x}{x^2} = \tfrac\pi2$ ; Gaussian $\int e^{-x^2} = \sqrt\pi$ and $\int_0^\infty e^{-x^2}\cos 2bx = \tfrac{\sqrt\pi}{2}e^{-b^2}$ ; King $\…; Formulas & Methods › Analysis
The six pools: ANALYSIS · LINEAR ALGEBRA · ALGEBRA & POLYNOMIALS · NUMBER THEORY · COMBINATORICS & PROBABILITY · GEOMETRY.; Everything the Putnam Tests › The Syllabus, Honestly
The squaring and reciprocal tricks: For $a_{n+1} = a_n + \tfrac{1}{a_n}$ , square: $a_{n+1}^2 = a_n^2 + 2 + a_n^{-2}$ , so $a_n^2 \approx 2n$ .; Hard Sequences: Closed Forms, Rates & Periodicity › Sequences the Putnam Actually Sets
The substitution playbook: Plug in $0$ , $1$ , $-x$ , $\tfrac1x$ , $1 - x$ , $y = x$ ; use the equation to build a second equation, then eliminate.; Functional Equations & Recurrences › Solving for Functions
The three series to know by sight: $\displaystyle\sum_{n\ge0}x^n = \frac{1}{1-x}$ , $\quad\sum_{n\ge1}\frac{x^n}{n} = -\ln(1-x)$ , $\quad\sum_{n\ge0}\frac{x^n}{n!} = e^x$ — and everything you get by differentiating or integrating them.; Sequences & Series › Summing and Testing
The universal problem-solving loop: Read twice.; Formulas & Methods › Combinatorics, Probability & Geometry
The words of the question are a contract: "Find all" means: every answer you list works, AND nothing else does.; Everything the Putnam Tests › Reading the Problem: What the Question Commits You To
Triage in the first ten minutes: Read all six problems.; Strategy & Writing a 10 › How to Take the Exam
Two kinds of improper, and the rule for both: An integral is improper at an infinite endpoint or at a point where the integrand blows up.; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › Improper Integrals: When They Converge
Two ways to say "there exists": Either CONSTRUCT the object, or TRAP it: the intermediate value theorem (a continuous function that is $\le 0$ at one point and $\ge 0$ at another vanishes between them), the pigeonhole principle ( $n + 1$ objects in $n$ boxes), or averaging (some term is at…; Proof Techniques for the Putnam › Existence: Construct It or Trap It
Uniqueness by integrating factor, and Grönwall: If $f' = f$ then $(fe^{-x})' = 0$ , so $f = Ce^x$ .; Differential Equations on the Putnam › The ODE Toolkit the Exam Assumes
Valuations: Legendre, Kummer, LTE: $v_p(n!) = \sum_{k\ge1}\lfloor n/p^k\rfloor$ ; $v_p\binom{n}{k}$ is the number of carries adding $k + (n-k)$ in base $p$ ; $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ for odd $p \mid a - b$ .; Number Theory › Proof-Level Number Theory
Vandermonde, circulant, tridiagonal: Vandermonde: $\det[x_i^{j-1}] = \prod_{i<j}(x_j - x_i)$ .; Linear Algebra I: Determinants & Structured Matrices › Determinants the Putnam Way
Vieta and Newton's identities: For $x^n - e_1x^{n-1} + e_2x^{n-2} - \cdots$ , the $e_k$ are the elementary symmetric functions of the roots, and the power sums obey $p_k = e_1p_{k-1} - e_2p_{k-2} + \cdots \pm ke_k$ .; Polynomials › Working With Roots You Never Find
Vieta jumping: For an equation quadratic in each variable, take a solution with minimal $x + y$ ; Vieta gives a second root, producing a smaller solution unless the minimal one is a fixed base case.; Extremal Principle and Working Backwards › The Extremal Principle
Weierstrass substitution and series expansion: $t = \tan\tfrac x2$ rationalizes any $\int R(\sin x, \cos x)\,dx$ .; Integrals III: Frullani, Beta, Gamma, Weierstrass & Series › The Named Integrals
Well-ordering as a proof engine: To prove a statement about objects measured by a nonnegative integer (a number, a degree, a sum $|x| + |y| + |z|$ ), assume a counterexample exists and take one with the SMALLEST measure.; Proof Techniques for the Putnam › Minimal Counterexamples, Worked
What a 10 looks like: A complete argument with every nontrivial step justified, edge cases handled, and the conclusion stated.; Strategy & Writing a 10 › How to Take the Exam
What a grader checks: A 10-point solution states what it proves, names every theorem it uses at the moment it uses it, checks that theorem's hypotheses, justifies every interchange of limits, handles every case (including boundary points and small $n$ ), and ends with the conclus…; Strategy & Writing a 10 › A 10 and a 2: Writing It Up
What an $\varepsilon$ - $\delta$ proof must contain: Given $\varepsilon > 0$ , EXHIBIT a $\delta > 0$ (or an $N$ ) and show the required inequality for every point within $\delta$ (or every index past $N$ ).; Limits, Continuity & the Big Theorems › Rigor: $\varepsilon$ - $\delta$ , Uniform Convergence and Compactness
What the exam is: Twelve problems, two three-hour sessions of six, each scored $0$ – $10$ .; Everything the Putnam Tests › The Syllabus, Honestly
Working backwards: Start from what you want and ask what would suffice.; Extremal Principle and Working Backwards › The Extremal Principle