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There are many guides on how to prepare for the Putnam, and so many people participate in the Putnam. However scoring perfect scores or anything reasonably close to a perfect score on the Putnam is so rare. It seems to me that the present guides make the basic assumption that answering everything on the Putnam is impossible, thus people who follow those guides would end up scoring 'normal' scores, just like everyone else. So, what specific steps should one take in order to get a perfect score on the Putnam? Is it important to master all of undergraduate mathematics and some graduate mathematics? Or is it really just a matter of practicing endless practice problems, just like all the other participants who don't get anything close to a perfect score?

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marked as duplicate by Zev Chonoles, Micah, Amzoti, Tom Oldfield, vonbrand May 24 '13 at 0:17

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You have to be fast. So not only should you practice lots of problems, you need to practice the art of guessing what the correct approach is from the outset. So it's basically all experience. Though, to actually get perfect on the Putnam, you'd probably need to also have the innate talent to hold lots of mathematical steps together in your head without resorting to having to write everything out, since that would waste too much time. –  Christopher A. Wong May 23 '13 at 20:42
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The premise of this question seems flawed. I don't think there's a secret way to score well on the Putnam that no one has discovered. You really do just have to practice a lot, have some talent, and get lucky. –  Potato May 23 '13 at 20:54
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I think this question should close, but the OP asks least one question we can answer concretely here. No, mastering all of undergraduate and graduate mathematics is not necessary. The Putnam is meant to be solvable for anyone who knows basic calculus up to multivariable, group theory, and linear algebra. Knowing some analysis and combinatorics and such goes a long way, but more abstract topics like topology, functional analysis, or ring theory will at best be tangentially relevant. –  neuguy May 23 '13 at 21:24
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@proximal I'm not sure that's entirely true. While the basic undergraduate curriculum will get you most of the way, in order to get a perfect score, it seems more is needed. I believe one of the recent tests had a question that ended up involving Fourier analysis on finite groups, for example. There was also a question on the 2011 exam where knowledge of $L^p$ spaces was helpful. –  Potato May 23 '13 at 21:32

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There is no useful advice, because the grading rubric on the Putnam makes it difficult to gain complete marks for an essentially correct solution. It is not all that unrealistic for some of the top competitors to have solved the important parts of every question, but to gain maximum points they need also to have checked every boundary case or small $n$, and not made some minor errors. You would need a combination of extreme preparation, tremendous speed, checking and re-checking of details, and still depend on luck to deliver up an exam "easy" enough (in the year to year variation) to be able to really solve everything. The last is probably the limiting factor, as there are enough multiple IMO gold winners taking the test now, every year, and an effectively unlimited amount of preparation material.

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A related question that must have a relatively definite answer is how much the grading allows quotation of theorems without proving them during the exam. For some of those problems, a known lemma from harmonic analysis or representation theory can accomplish significant progress within a solution, but I think is not accepted as a full answer. –  zyx May 23 '13 at 23:13
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I think there was a year, for example, where a well known expert took the test, whose level of ability and preparation would be probably impossible for most olympiad winners to replicate (given unlimited time and optimal conditions). He got something like 117 of the maximum 120, and the next highest score was 30 points lower. I would guess that the subtraction of points was not from failing to solve any problem, but minor errors or omitting proofs of some facts used in the solutions. –  zyx May 23 '13 at 23:28

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