What the Putnam Actually Is
The William Lowell Putnam Mathematical Competition is an annual math contest for undergraduate students in the United States and Canada. It runs on the first Saturday of December and consists of two sessions, three hours each, with six problems per session. The scoring is out of 120 points. Few students score above 30. The problems are not computational drills. They require insight, cleverness, and the ability to construct rigorous arguments under time pressure. People often assume the Putnam tests advanced graduate-level mathematics. It doesn't. The problems are drawn from first-year calculus through real analysis, linear algebra, combinatorics, and number theory. The difficulty comes from how the problems are framed, not from the level of mathematics required. You need to know definitions and theorems correctly. But more importantly, you need to recognize what structure is hiding behind an ugly statement.
William Lowell Putnam Mathematical Competition Problems And Solutions
This is the primary resource most people use when preparing. The official problems and solutions have been published annually since 1938. The website for the competition itself lists every past paper with full solutions. You can find them organized by year going back decades. Some third-party sites compile solutions written by individual competitors rather than the official panels. Those are worth reading because they sometimes show multiple approaches. But the official solutions are the gold standard for understanding what a complete answer looks like. Putnam problems reward clean thinking and punish brute force. A typical problem might ask you to evaluate an integral, prove a bound, or find all functions satisfying a given equation. The path to the answer is rarely obvious on first read. I remember working through the 2004 B6 problem, which asked for the maximum number of acute triangles formed by vertices of a regular polygon. I spent forty minutes trying coordinate geometry and got nowhere. The trick was to use an angle-chasing argument paired with a parity observation about how many angles could be acute at each vertex. Once you see that, the solution is three lines. Before you see it, it looks impossible. That pattern repeats across almost every problem set. The competition tests whether you can shift perspectives quickly. Computational ability matters, but secondary. The real skill is knowing which tool to reach for and when to abandon the tool you are currently using.
Where to Find the Problems and Solutions
The official source is the Putnam exam page hosted by the Mathematical Association of America. They archive every year from 1938 onward. Each entry contains the full problem set and the official solution. The MAA also publishes a yearly volume called The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary edited by Korevaar and Sarkozy. That book is widely considered the best single-volume resource available. It includes commentary alongside solutions, which explains the motivation behind key steps rather than just presenting a finished proof. For free online access, search for the MAA Putnam archive directly. Avoid sites that require registration or push subscriptions. The problems are public domain in practice, and the solutions are published by the MAA. If a site is asking you to create an account to read a problem from 1997, you are looking at the wrong place.
Get the Full Details
How to Use the Solutions Effectively
Reading a solution after giving up on a problem is useful but limited. The more valuable practice is attempting the problem for at least forty-five minutes before looking anything up. If you cannot make progress, spend another thirty minutes writing down every relevant definition, theorem, and special case you can think of. Then check the solution. Afterward, close the solution and re-derive it from memory on blank paper. This forces you to reconstruct the logical flow rather than passively recognizing steps. I used to skip the re-derivation step because it felt redundant. That changed when I realized I could follow a solution during reading but consistently fail to reproduce the key substitution when solving independently. The gap between recognition and production is where most students lose points. Re-deriving closes that gap.
Common Pitfalls
Students tend to over-index on learning new theorems instead of deepening their grasp of familiar ones. Knowing the statement of the intermediate value theorem is not the same as knowing how to apply it in a situation where the function is defined piecewise or implicitly. The Putnam frequently uses results from analysis in contexts where the conditions are disguised. You need to be comfortable verifying hypotheses, not just recalling conclusions. Another frequent mistake is treating combinatorics problems as calculation problems. Many Putnam combinatorics questions can be solved with a clean bijection or double-counting argument. If you are setting up summations with binomial coefficients, pause and ask whether there is a structural reason the count should equal something simpler. I once spent twenty minutes expanding a sum involving terms before realizing the problem was asking for the number of subsets with even cardinality, which is exactly . The expansion was correct. It was just the wrong path.
Limitations of This Preparation Method
Working through past problems and solutions is the most effective preparation strategy available. It is not sufficient on its own. The Putnam rewards a certain type of mathematical maturity that comes from sustained engagement with difficult problems over months or years, not from cramming in the weeks before December. Students who treat it as a four-week study project typically see minimal improvement regardless of how many solutions they read. Additionally, the official solutions are sometimes brief. They state the key insight and then move forward. Reading them can create a false sense of understanding. You recognize the argument and think you could replicate it. You cannot, until you have actually struggled with similar problems independently. Supplement official solutions with books like Putnam and Beyond by Andreescu and Gelca, which provides more expansive explanations and connects problems to broader mathematical ideas.
A Practical Study Sequence
Start with problems from the 1980s and 1990s. The style is slightly more accessible than the most recent sets, and solutions are well-documented. Work A and B problems separately. A problems generally require less machinery. B problems tend to be harder but not always deeper. Attempt each problem for a fixed period before checking the solution. Record which types of problems you consistently miss. Build a targeted review plan around those categories rather than randomly solving new problems. The competition has been running since 1938. There are more than eighty years of problems available. That is enough material to sustain serious preparation through the undergraduate years. The resource is there. The question is whether the practice matches the demand the exam places on you.
