Resource guide
Putnam Study Resources
Books, archives, and topic-by-topic advice for studying the Putnam. Built for Tulane students who've never done competition math before, and useful for people who have.
How to use this page
If you've never studied for Putnam before
The Putnam isn't a test of how much math you've taken; it's a test of how you think about problems. The median score is famously 0 or 1 out of 120. That's not a joke about how hard it is. It's a sign that the people who do well aren't the ones who've memorized the most theorems; they're the ones who've seen the most problems and recognize patterns.
The cheapest, highest-yield thing you can do is read past problems and their solutions. Even when you can't solve one, reading a clean solution and asking "what would I have needed to notice?" teaches you the pattern. Do that fifty times and the next A1 looks familiar.
Quick orientation
- Format: Two 3-hour sessions (A and B), 6 problems each, scored 0 to 10 per problem.
- A1 / B1: The easiest problem on each session. Most accessible to first-timers.
- A6 / B6: Brutally hard. Skip them at first.
- Tone: Short, elegant problems. Cleverness over computation.
Books
What to read
You don't need to own all of these. Pick one for technique and one for problems.
- All levels
Putnam and Beyond
Răzvan Gelca, Titu Andreescu
The most comprehensive single book. ~800 problems organized by topic with full solutions. The standard reference for serious prep.
- Intermediate
Problem-Solving Strategies
Arthur Engel
Olympiad-flavored but the techniques (invariants, extremal principle, coloring) transfer directly to Putnam. Great for building intuition.
- Beginner
The Art of Problem Solving, Volume 2
Sandor Lehoczky, Richard Rusczyk
Reads like a friendly mentor. Best on-ramp if you have no contest background. Skip Vol. 1 if you've taken Calc I.
- Intermediate
Mathematical Olympiad Challenges
Titu Andreescu, Răzvan Gelca
Problems grouped by trick. Useful for drilling a single technique at a time.
- All levels
The William Lowell Putnam Mathematical Competition (1985–2000)
Kedlaya, Poonen, Vakil
Every problem from 1985 to 2000 with multiple solutions and historical notes. Worth owning.
- All levels
How to Solve It
George Pólya
Not a problem book. The book on mathematical problem solving. Short, old, still the best 80 pages on the topic.
The Tulane Math Department library has copies of most of these. Email an officer if you can't find one.
Problem archives
Where to find past problems
Old problems are the curriculum. These four cover everything you need.
Putnam Archive (Kiran Kedlaya)
The definitive archive. Every Putnam problem and solution from 1985 onward, maintained by a former Putnam Fellow.
Art of Problem Solving: Putnam Forum
Active discussion of each year's problems. Often has multiple elegant solutions per problem.
Putnam Competition (official MAA)
Official rules, registration, and recent winners. Where Tulane registers students each year.
AoPS Wiki: Putnam
Yearly problem indexes with links to discussion threads. Easy to browse.
Topics
What actually shows up
Putnam problems cluster around a handful of topics. Drill one at a time.
Number theory
Roughly two problems per exam. Heavy on modular arithmetic, divisibility, primes, and Fermat / Euler.
Drill these
- Fermat's little theorem and Euler's theorem (and when each applies)
- Chinese Remainder Theorem and lifting the exponent
- Order of an element mod n; primitive roots
- p-adic valuation and 2-adic tricks
Combinatorics
Counting, pigeonhole, generating functions, and graph-coloring arguments. Often dressed up in unfamiliar settings.
Drill these
- Bijective proofs and double counting
- Pigeonhole and extremal principle
- Generating functions for closed forms
- Inclusion-exclusion on small cases
Analysis & calculus
The A1 / B1 problems often look like cleverly-disguised calculus questions. Real analysis intuition pays off later.
Drill these
- Series convergence tests and tricky sums
- Mean value theorem, intermediate value theorem
- Suprema, limits, and ε-N estimates
- Integration tricks (symmetry, substitution, differentiation under the integral)
Linear algebra
Eigenvalues, determinants, and rank arguments. Surprisingly versatile on combinatorial problems too.
Drill these
- Determinant identities and the Cauchy-Binet formula
- Trace tricks (especially tr(AB) = tr(BA))
- Characteristic and minimal polynomial
- Rank-nullity in disguise
Inequalities & algebra
AM-GM, Cauchy-Schwarz, and clever substitutions. The easiest topic to drill, and high yield on A1 / B1.
Drill these
- AM-GM and weighted AM-GM
- Cauchy-Schwarz in Engel (Titu) form
- Power mean inequality
- Smoothing / tangent line tricks
Geometry
Less common than the others, but worth a week. Often blends with linear algebra or complex numbers.
Drill these
- Complex numbers for plane geometry
- Vectors, dot products, cross products
- Affine transformations (and when they preserve the problem)
- Classic theorems: Ptolemy, Ceva, Menelaus
A starter plan
Your first month
If you're starting from zero, here's a low-effort, four-week ramp.
- Week 1
Read one A1 and one B1 per day
A1 and B1 are the easiest problem on each session. Twenty minutes each, no looking up. Write down whatever you observe, even if you don't solve it.
- Week 2
Pick one topic and drill it
Choose number theory or inequalities (highest yield for beginners). Work through a chapter of Putnam and Beyond on that topic.
- Week 3
Take a mock half-Putnam
Three hours, six problems from one session of a recent year. No interruptions. Then go through every problem afterward with the solution.
- Week 4
Come to prep, bring your write-ups
The fastest way to improve is to talk through your reasoning with someone else. That's what the weekly Putnam prep session is for.
Come to a prep session
Weekly, friendly, beginners welcome. The fastest way to make progress is to work through problems with other students.