TITLE: Studying for the Putnam Exam
QUESTION [134 upvotes]: 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.)

REPLY [23 votes]: I took the Putnam four times, $1965$-$1968$. I never knew my scores, but my ranks, if I remember correctly, were in the $80$s my first year, abysmal my second year — in the $700$s, I think — and somewhere between $100$ and $150$ my last two years. (My background and interests just weren’t well-suited to that $1966$ exam; the other three exams suited me much better.) That’s hardly a spectacular Putnam career, but it’s certainly respectable.
I never did anything specifically to prepare for for the exam: I was simply taking it for fun. 
However, I’d known since before I was ten that I was going to be a mathematician, teaching mathematics at the college/university level. By the time I actually went off to college, I’d completed standard college courses in calculus, advanced calculus, differential equations, and complex variables. I’d also had a course in elementary number theory and another combining bits of group theory and linear algebra in the Ross Mathematics Program, then at Notre Dame, the summer after my first year of high school. It’s perfectly true, therefore, that I’d had a lot of exposure to mathematics, including writing it, before I first took the Putnam. At the same time I should point out that I had quite a few other interests and by no means spent all my time on mathematics and closely related subjects.
A couple of points come to mind that seem worth mentioning. That was a different era: there was nothing like the number of mathematics competitions that are available today, and there was less that one could easily do to prepare for the Putnam. Nowadays it’s much easier to gather competition experience, if one has the talent and interest, and I’ve a strong suspicion that such experience confers a significant advantage. It’s also much more common now for a mathematics department to run problem-solving seminars, or even courses. On the other hand, I would not have been much interested in developing specifically competition-oriented skills even if I’d had the opportunity: I enjoy solving problems that I find interesting, and a lot of the standard competition tricks and topics simply don’t much interest me and never did. I’d have considered my time much better spent learning more ‘real’ mathematics.