Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career trajectories of former participants, or other such things. This is a question about how one can most effectively prepare to do well.

Many such questions have been asked before on this site. The answers, while helpful, all seem to boil down to recommendations of the same standard canon of books with the encouragement to do more problems. This is very good advice, but I want to approach the topic from a different angle.

We have a wealth of talent on this website, and in particular many users who have done quite well at these competitions. I would like to hear their personal stories. In particular, how did you prepare, and what kind of time commitment did you put forth? And advice, especially practical study tips, is always welcome. :)

edit: Also, it seems to me that the majority of people who do very well at the Putnam had developed the majority of their skill in high school, and focused mainly on their classes in college (while attending whatever Putnam seminar their college offered). I would appreciate comments on this matter too.

(Motivation: It seems to me like the role of talent is vastly overestimated in mathematics, and in mathematics competitions in particular, to the point where the Putnam exam gets used as a sort of pseudo-IQ test. Of course the people who do well have gifts, but it also seems that, without fail, they all have a history of doing many hours of mathematics a day for years on end. For example, I recall reading an interview with Tao where he admitted his childhood consisted of nothing but math and computer games. I am trying to gather some evidence on this matter [and advice for Putnam preparation!]. Please keep the answers focused on the actual question, though.)

I definitely think the Putnam exam tests mathematical "maturity" just as much as talent or raw knowledge. By maturity, I mean the somewhat less tangible things we pick up like gauging a problem's difficulty, anticipating the necessary techniques, knowing how to construct an intelligible argument, and knowing when to move on. Ego can be a huge issue as well - I know it was for me. I missed a lot of points on my second try because I tried to solve too many of the problems and ended up with many incomplete solutions instead of a couple complete ones.

I took the exam twice - once as a sophomore barely out of multivariable calculus, and again as a senior. While I had certainly learned more "math" in those two years, I think the real difference was that I had learned how to make a valid mathematical argument. So my 300-level intro to proof writing and analysis class was much more important than my 400-level differential equations and probability classes. I think the only thing that prevented me from doing extremely well was my ego (and a bit of experience).

As far as practicing/studying/preparing, my department offered a "prep" class which was taught by a former IMO medalist. We worked problems, mostly, but he was able to offer a lot of good strategies and tricks as well. The biggest takeaway message seemed to be that preparing to solve a problem is just as important as actually solving it! I.e. taking time to decompose the problem into its core pieces, look for tricks or "dimension reductions," try to figure out what the question is "really" about (i.e. maybe its not a linear algebra problem, but a counting problem). While you can sometimes make progress by jumping in the deep end, a little big-picture can go a long way. This is in stark contrast to the math GRE!

I also can't recommend enough cross-training in the form of reading about problem solving. Books like "How to Solve It", "Proofs from The Book", and blog posts by Terry Tao, etc. are a great resource. Books in a similar vein from the "other" sciences, like Feynman, can be surprisingly helpful as well - the techniques of discovery.

The people who did well on the Putnam tend to have experience from high school.

My sophomore year in college, I answered 3 questions on the test. That was enough to be significant, and I got invitations to special summer programs, like Budapest Semsters in Mathematics and Math in Moscow.

Titu Andresscu has a nice book Putnam and Beyond which gives an alternative look at the standard undergraduate curriculum. MIT used to offer a problem solving seminar. Arthur Engel's Problem Solving Strategies is nice.

Personally, I think you shouldn't feel guilty about looking at the answers. I don't know how you're supposed to guess the answer if you've never seen these problems before. Sometimes, the answers may be correct but don't really explain what's happening. So you have to read them critically, get some context.

Keep in mind, the Putnam is a writing contest, so the proofs have to be written correctly. The graders are rough and don't give credit if not all the steps are included. Maybe Terence Tao's collection of math writing articles will help here.