I began collecting these resources in 2022, when I was an undergraduate without much mathematical competition experience. My main advice is to combine reading with sustained attempts at problems and careful proof writing. Completing a long book series is not a prerequisite for trying a Putnam problem.
The MAA's Putnam page is the place to check current participation rules and competition arrangements. Its archive of past competitions provides problems and solutions for practice.
A workable practice cycle
Choose a few problems that you can understand even if you cannot yet solve them. Earlier problems such as A1 and B1 are reasonable starting points, although difficulty varies by year.
- Work independently for a sustained period. Record examples, partial deductions, and failed approaches.
- If stuck, read a hint or the beginning of a solution, then try again before reading the rest.
- Write a complete proof in your own words. Check the quantifiers, boundary cases, and any division by an expression that might be zero.
- Revisit the problem later without the solution. Identify the idea you could reuse on a different problem.
Mix this slower work with occasional timed sessions. Afterward, separate gaps in background knowledge from difficulty finding an approach or communicating a proof. Those problems call for different kinds of practice.
Books I considered
For readers comfortable with Chinese, I originally recommended 《数学奥林匹克小丛书》 as a source of topic-based foundations and 《数学奥林匹克命题人讲座》 for more advanced competition work. I was reading the former when I wrote the first version of this note.
Use individual volumes to address specific gaps. An undergraduate need not be embarrassed to study a middle-school topic, but also need not complete every middle-school and high-school volume before moving on. Olympiad material helps with algebra, combinatorics, inequalities, and number theory; Putnam preparation also benefits from undergraduate analysis and linear algebra.
For additional practice collections and advice, see Hildebrand's Putnam resources at Illinois and David J. Wright’s Power Putnam Preparation. Choose resources whose solutions explain the reasoning at a level you can follow, then spend enough time using them to assess whether they help.
Adjust the plan without judging your ability from one result
A difficult chapter or a low competition score does not establish a fixed limit on mathematical ability. Try a more accessible problem set, get feedback on your proofs, or reduce the amount of new material studied at once.
It is also reasonable to spend less time on competitions if your interests lie elsewhere. Competition problem solving is one part of mathematics, and preparation should serve your interests rather than become a test of whether you belong in the subject.