Putnam Solutions Guide

Each December, college students in the United States and Canada sit the William Lowell Putnam Mathematical Competition: problems A1 to A6 in the morning and B1 to B6 in the afternoon, three hours each, with the later problems usually the hardest. Here, year by year, are short topic labels, a link to each official problem, and our own solutions.

About these pages. The Putnam problems are © the Mathematical Association of America. We don't copy them: read each one at the official link, on Kiran Kedlaya's Putnam Archive, which hosts them with the MAA's permission. The hints and solutions are our own work, written from scratch. They are not official solutions.

Years

Every problem

Problem	Topic	Area	Level	Solution
2025 A1	Iterating a fraction map: eventually coprime numerators	Number theory	2	Ready
2025 A2	Best parabola bounds for the sine on one arch	Calculus, Analysis	2	Ready
2025 A3	Game: walking on ternary strings without revisiting	Combinatorics	3	Ready
2025 A4	Smallest matrices whose commuting pattern is a cycle	Linear algebra	3	Ready
2025 A5	Which up-down patterns allow the most permutations?	Combinatorics	5	Ready
2025 A6	2-adic behavior of a quadratic recursion	Number theory	4	Ready
2025 B1	Two-colored plane closed under circumcenters	Geometry, Combinatorics	2	Ready
2025 B2	Centroids of a region and of its solid of revolution	Calculus, Analysis	1	Ready
2025 B3	Sets closed under divisors of a difference of powers	Number theory	3	Ready
2025 B4	Sum versus support of a slowly rising staircase matrix	Combinatorics, Algebra	4	Ready
2025 B5	Descents in the sequence of modular inverses	Number theory, Combinatorics	3	Ready
2025 B6	Fastest-growing exponent for a self-referential difference inequality	Analysis, Number theory	5	Ready
2024 A1	When can twice, thrice and four times powers balance?	Number theory	2	Ready
2024 A2	Polynomials whose second iterate is divisible by a square	Algebra	2	Ready
2024 A3	Probability inside random three-row standard tableaux	Combinatorics, Probability	3	Ready
2024 A4	Geometric sequences that rearrange into arithmetic ones mod p	Number theory	3	Ready
2024 A5	Probability: random chords avoiding an off-center disk	Probability, Geometry, Analysis	4	Ready
2024 A6	Hankel determinant of a quadratic generating function	Linear algebra, Combinatorics, Algebra	4	Ready
2024 B1	Choosing a permutation of cells with consecutive labels	Combinatorics	1	Ready
2024 B2	Partner quadrilaterals and congruence classes	Geometry	3	Ready
2024 B3	Gaps between consecutive solutions of tan x = x	Analysis, Calculus	2	Ready
2024 B4	Expected position of a walk drifting toward the middle	Probability	2	Ready
2024 B5	Counting pairs of chains bounded by a common top	Combinatorics, Algebra	5	Ready
2024 B6	Growth of a lacunary series near the unit circle	Analysis	4	Ready
2023 A1	Second derivative of a product of cosines	Calculus	1	Ready
2023 A2	A monic polynomial matching reciprocal points	Algebra	2	Ready
2023 A3	How soon can a function controlled by its partner vanish?	Analysis	3	Ready
2023 A4	Integer combinations of icosahedron vertices are dense	Geometry, Number theory	2	Ready
2023 A5	Roots of a sum weighted by ternary digits	Algebra, Combinatorics	4	Ready
2023 A6	Parity game on chosen-number fixed points	Combinatorics	5	Ready
2023 B1	Sliding coins on a grid: counting reachable positions	Combinatorics	2	Ready
2023 B2	Fewest binary ones in a multiple of 2023	Number theory	2	Ready
2023 B3	Expected longest zigzag subsequence of random numbers	Probability, Combinatorics	3	Ready
2023 B4	Optimizing a piecewise-accelerating function to reach 2023	Calculus, Analysis	3	Ready
2023 B5	Square roots of multiplication permutations modulo n	Number theory, Combinatorics	4	Ready
2023 B6	Determinant of a matrix counting linear representations	Linear algebra, Combinatorics	5	Ready
2022 A1	Lines meeting a logarithmic curve exactly once	Calculus, Analysis	2	Ready
2022 A2	Negative coefficients in the square of a polynomial	Algebra	3	Ready
2022 A3	Counting periodic sequences of a five-cycle recurrence	Number theory, Combinatorics	3	Ready
2022 A4	Probability: weighted sum until the first rise	Probability	2	Ready
2022 A5	Domino-placing game on a long strip	Combinatorics	5	Ready
2022 A6	Intervals with equal odd-power length sums	Analysis, Algebra	5	Ready
2022 B1	Exponential of an integer polynomial: no zero coefficients	Algebra, Number theory	2	Ready
2022 B2	Finite sets closed under the cross product	Linear algebra, Geometry	2	Ready
2022 B3	Recoloring the positive reals by same-color distances	Combinatorics	3	Ready
2022 B4	Cyclic sequences whose consecutive triples are progressions	Combinatorics, Number theory	4	Ready
2022 B5	Probability: when zero is the likeliest weighted sum	Probability, Analysis	4	Ready
2022 B6	A functional equation for positive continuous functions	Analysis, Algebra	5	Ready
2021 A1	Lattice hops of length 5 toward a far point	Number theory, Combinatorics	1	Ready
2021 A2	A limit of a power expression as the exponent vanishes	Analysis, Calculus	1	Ready
2021 A3	Integer regular tetrahedra inscribed in a sphere	Geometry, Number theory, Linear algebra	3	Ready
2021 A4	A plane integral that collapses onto an annulus	Analysis, Calculus	4	Ready
2021 A5	Power sums of units modulo 43 times 47	Number theory	2	Ready
2021 A6	Reducible 0–1 polynomials and composite values at 2	Algebra, Number theory	4	Ready
2021 B1	Probability: a randomly dropped square avoids grid corners	Probability, Geometry	2	Ready
2021 B2	Inequality: weighted geometric means of a summable sequence	Analysis, Algebra	3	Ready
2021 B3	Rotation derivative integrated over a moving disk	Analysis, Calculus	2	Ready
2021 B4	Fibonacci moduli: a product of self-powers	Number theory	4	Ready
2021 B5	Matrices with every principal minor odd	Linear algebra, Combinatorics	4	Ready
2021 B6	Probability: repeated median-of-three trimming	Probability, Analysis	3	Ready
2020 A1	Number theory: repunits followed by zeros, divisible by 2020	Number theory	1	Ready
2020 A2	Combinatorial identity: a coin-flipping race	Combinatorics, Probability	2	Ready
2020 A3	Series: iterating sine slowly	Analysis	2	Ready
2020 A4	Probability: random blackening on a strip, expected leftovers	Probability, Combinatorics	3	Ready
2020 A5	Fibonacci representations: the last number with exactly 2020	Combinatorics, Number theory	4	Ready
2020 A6	Fourier series: Fejér means of a square wave	Analysis	4	Ready
2020 B1	Number theory: Thue–Morse signs kill low powers	Number theory, Combinatorics	2	Ready
2020 B2	Games: moving pegs right, a pairing strategy	Combinatorics	3	Ready
2020 B3	Probability: shrinking intervals and a Poisson count	Probability	2	Ready
2020 B4	Lattice paths: averaging over cyclic rotations	Combinatorics, Probability	3	Ready
2020 B5	Complex numbers: inversion turns the unit circle into a line	Algebra, Geometry	4	Ready
2020 B6	Number theory: signs from a Beatty sequence	Number theory	5	Ready
2019 A1	Number theory: values of a classic cubic form	Number theory, Algebra	1	Ready
2019 A2	Triangle geometry: incenter, centroid and a half-angle	Geometry	2	Ready
2019 A3	Polynomials: average root size with increasing coefficients	Algebra, Analysis	3	Ready
2019 A4	Analysis: functions with zero average on unit spheres	Analysis, Calculus	3	Ready
2019 A5	Finite fields: multiplicity of the root 1	Algebra, Number theory	4	Ready
2019 A6	Analysis: flatness at zero versus second-derivative growth	Analysis, Calculus	4	Ready
2019 B1	Lattice geometry: squares among points on dyadic circles	Combinatorics, Geometry	2	Ready
2019 B2	Trigonometric sums: a telescoping identity	Analysis	2	Ready
2019 B3	Linear algebra: a reflection times an orthogonal matrix	Linear algebra	2	Ready
2019 B4	PDEs: two equations pin down the mixed partial	Analysis, Calculus	3	Ready
2019 B5	Interpolation: Fibonacci numbers and finite differences	Algebra, Combinatorics	3	Ready
2019 B6	Lattices: perfect codes in every dimension	Combinatorics, Number theory	3	Ready
2018 A1	Number theory: unit fractions and a factoring trick	Number theory	1	Ready
2018 A2	Linear algebra: the intersecting-subsets matrix	Linear algebra, Combinatorics	3	Ready
2018 A3	Optimization: sums of cubes with zero sum	Algebra, Analysis	3	Ready
2018 A4	Group theory: Euclid’s algorithm on a balanced word	Algebra, Number theory	4	Ready
2018 A5	Analysis: a smooth function must have a negative derivative	Analysis, Calculus	3	Ready
2018 A6	Geometry: rational squared distances force rational area ratios	Geometry, Algebra	3	Ready
2018 B1	Combinatorics: removing a vector to allow a balanced split	Combinatorics	2	Ready
2018 B2	Complex analysis: a polynomial with no roots in the disk	Algebra, Analysis	2	Ready
2018 B3	Number theory: towers of powers of two	Number theory	2	Ready
2018 B4	Recursions: Chebyshev polynomials at Fibonacci indices	Algebra, Number theory	3	Ready
2018 B5	Calculus: a positive-definite Jacobian forces injectivity	Analysis, Linear algebra	3	Ready
2018 B6	Counting: a generating-function bound on sequences	Combinatorics, Analysis	3	Ready
2017 A1	Number theory: closure rules with squares and residues	Number theory	2	Ready
2017 A2	Polynomials: a nonlinear recursion that is secretly linear	Algebra	2	Ready
2017 A3	Integrals: ratios of powers grow without bound	Analysis	2	Ready
2017 A4	Combinatorics: splitting a class into equal-average halves	Combinatorics	3	Ready
2017 A5	Probability: a card game and cycles of permutations	Probability, Combinatorics	4	Ready
2017 A6	Coloring: icosahedron edges, counting with arithmetic mod 3	Combinatorics, Linear algebra	4	Ready
2017 B1	Geometry: lines, homotheties and parallelism	Geometry	1	Ready
2017 B2	Number theory: sums of consecutive integers and odd divisors	Number theory	2	Ready
2017 B3	Power series with 0/1 coefficients: an irrationality proof	Number theory, Analysis	3	Ready
2017 B4	Series: logarithms with a period-four pattern	Analysis	3	Ready
2017 B5	Geometry: lines halving area and perimeter of triangles	Geometry	4	Ready
2017 B6	Counting: distinct tuples with a weighted sum mod a prime	Combinatorics, Number theory	5	Ready
2016 A1	Number theory: derivatives of integer polynomials and factorials	Number theory, Algebra	1	Ready
2016 A2	Asymptotics: comparing two binomial coefficients	Combinatorics, Analysis	2	Ready
2016 A3	Functional equation: a three-cycle and arctangent integrals	Analysis, Calculus	2	Ready
2016 A4	Tiling: odd rectangle with trominoes and skew tetrominoes	Combinatorics	3	Ready
2016 A5	Group theory: words in alternating generators	Algebra	3	Ready
2016 A6	Inequality: integral versus maximum of a cubic	Analysis	5	Ready
2016 B1	Series: a telescoping recursive sequence	Analysis	1	Ready
2016 B2	Counting: numbers near perfect squares, growth rate	Number theory, Analysis	2	Ready
2016 B3	Geometry: covering points with a bounded triangle	Geometry	2	Ready
2016 B4	Probability: expected determinant of a skew-symmetric matrix	Probability, Linear algebra	3	Ready
2016 B5	Functional inequality: maps respecting power intervals	Analysis	4	Ready
2016 B6	Series: a double sum that telescopes by halving	Analysis	4	Ready
1992 A1	Functional equations: an involution on the integers	Algebra	1	Ready
1992 A2	Calculus: a binomial coefficient times a logarithmic derivative	Calculus, Combinatorics	2	Ready
1992 A3	Diophantine equations: sums of squares versus products	Number theory	3	Ready
1992 A4	Analysis: smooth functions fixed by values on 1/n	Analysis, Calculus	2	Ready
1992 A5	Combinatorics on words: the Thue–Morse sequence has no cubes	Combinatorics	4	Ready
1992 A6	Probability: random points on a sphere	Probability, Geometry	3	Ready
1992 B1	Combinatorics: fewest distinct pairwise averages	Combinatorics	1	Ready
1992 B2	Polynomials: factoring a four-term sum	Combinatorics, Algebra	1	Ready
1992 B3	Dynamics: where a quadratic iteration converges	Analysis	3	Ready
1992 B4	Rational functions: high derivatives and cancelling leading terms	Algebra, Calculus	3	Ready
1992 B5	Determinants: a diagonal plus all-ones matrix	Linear algebra	2	Ready
1992 B6	Linear algebra: commuting-or-anticommuting matrix families	Linear algebra, Algebra	4	Ready

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