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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.)

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+1: for your final musings. "We excel at what we repeatedly do." (Aristotle) –  gnometorule Feb 16 '13 at 20:08
Maybe you can get some good responses also at the site artofproblemsolving.com –  GEdgar Feb 16 '13 at 20:25
'the role of talent is vastly overestimated in mathematics' There's this thing called deliberate practice, and this book called The Cambridge Handbook of Expertise and Expert Performance (I very highly recommend this book). It says many things which sound like 'doing deliberate practice 4-5 hours/day for many years makes a mathematician' and 'hard work (deliberate practice) makes an expert, not innate talent'. Btw deliberate practice is hard work but not all hard work is deliberate practice. I just wanted to tell you because you seem very interested in it. –  raindrop Aug 10 '13 at 1:02
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6 Answers

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 gold 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.

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I totally agree that the proof writing classes and the resulting understanding of making a logical argument starting from premises while defining every item along the way, is more valuable than just knowing how to do calculus!! –  Joseph Myers Apr 3 '13 at 23:44
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I took the Putnam exam twice and did much better the second time around. There were no special Putnam preparation courses, but I did take reading courses (both before the first Putnam attempt and between the two attempts) from a professor whose idea of "reading" was "struggling with problems". That undoubtedly helped, and I surely learned some more math during the year between my two attempts, but my subjective impression is that the real difference between the two is that I just happened to have a good day the second time around. So my advice, in addition to the obvious things about studying and practicing, is the even more obvious: Get plenty of sleep the night before.

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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.

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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.

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@JoelReyesNoche Yes, it's been edited. –  raindrop May 25 '13 at 3:24
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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).

The last statement is very much false for the top $n$ rankings for not too small $n$. What is more likely --- I know it to be true for many top 10 performances and have no reason to believe that most of the others are different --- is that people who make the highest ranks often focus on the Putnam and then complete that semester's schoolwork after the competition is over. And if that is impossible at their school, or there is no academic credit for a Putnam class or "independent research/problemsolving" seminar that allows coursework to help preparation instead of impeding it, then their ranking can suffer. One reason Harvard is less dominant in the Putnam contest today is that its academic calendar changed, so that early December is near the end of the term. In earlier days the term would end in mid- or late January, with a long study period that had no classes, so that catching up a semester of work was very feasible if you were a top performer on the Putnam and many of your classes were in mathematical subjects. There are other things that have changed, such as a large number of universities now recruiting math competition winners, holding Putnam seminars and awarding scholarships for mathematicians who can represent the school at contests, so that on the whole, the contest rankings are detecting preparation and school support more than they used to.

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I awarded you the bounty because I haven't heard this information anywhere before. Thanks. –  Potato Apr 4 '13 at 17:24
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I would suggest practicing with these previous tests, and only looking at the answers after spending six hours on each test. http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml

Also, get in the habit of reading theorems and absorbing what they can actually do by trying to imagine how you would use them and what kind of problem they would enable you to solve. That way it will be easier to make theorems a part of your intuition so that you can remember them.

Just trying to memorize theorems is actually both the worst way to remember them and the worst way to realize which theorem you can apply to a type of problem.

Building skill in solving unexpected problems, like those on Putnam exams, is very hard, and partly a matter of experience that gives you confidence. If you get good at it, though, you might earn some extra scholarship money (I don't know if you need that, but as a student I did, and I was very thankful for it).

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You may do that if you like. I did not know there were official solutions now available online. –  Joseph Myers Apr 4 '13 at 3:21
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