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Currently, my math training includes Calc 1-3, linear algebra, and some introduction to set theory/discrete math. What would you recommend that I study over summer in preparation for the Putnam? Real analysis, topology, abstract algebra (all of the above)? What would be the most pertinent? Thanks!

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Old contests! And just reading solutions is roughly equivalent to doing nothing. –  André Nicolas May 20 '11 at 0:56
    
@user6312: so you would say that even with my scant mathematical background I do not need to study further subjects? –  Aspirant May 20 '11 at 1:27
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Oh, you do, but not necessarily for the Putnam. Anyway, Putnam and other high level contests will lead you to look at relevant mathematics. There are also good Putnam preparation web sites, for example the one at Berkeley that used to be run by Professor Kahan. –  André Nicolas May 20 '11 at 1:32
    
@user6312: sorry that's what I meant. So instead of focusing on a particular subject (say, RA) I should instead go over old problems and see what's relevant? –  Aspirant May 20 '11 at 1:40
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There is also a nice MIT site, and there are others, like Virginia Tech if I recall. Oh yes, do look at Real Analysis, or other stuff, but not for the sake of the Putnam. Putnam eligibility only lasts for a little while, mathematics is forever. –  André Nicolas May 20 '11 at 1:49

3 Answers 3

There are lots of good books.

  • William Lowell Putnam Mathematical Competition: Problems & Soluti​ons: 1938-1964
  • William Lowell Putnam Mathematical Competition: Problems & Soluti​ons: 1965-1984
  • William Lowell Putnam Mathematical Competition: Problems & Soluti​ons: 1985-2000
  • Putnam and Beyond
  • Problem-Solving Strategies

The first 3 are all the contests from 1938 through 2000, and includes solutions to all of them. The last 2 have lots of problems that are arranged by topic. So, those are some topics you can study. The last 2 are probably better for this reason because it teaches you many strategies/problem solving techniques and gives you problems that you can try involving those techniques. Then, you can try other problems from the first 3 books where they aren't organized by topic. If you're in college now, your library may have some or all of these books, or similar books.

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Putnam does not require any knowledge of analysis/algebra/topology, just do past exams like suggested above by user6312.

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There are often problems on the Putnam that require a little bit of Group Theory. If you want to get a perfect score on the Putnam, you probably do have to know a little of those other topics. But if you're willing to settle for finishing in the top 100, you can do that with the topics you already know - that, and some good test-taking skills, and some luck. Oh, and if I were going to add one more subject, it would be Number Theory (but maybe you already did some in Discrete Math). –  Gerry Myerson May 20 '11 at 4:12

If you want to do well on the Putnam, I think you'd do well to look over the books that Graphth suggested. However, to dismiss real analysis, abstract algebra, and topology, could be a pretty grave mistake- there are often at least 2-3 questions that cover those topics.

Fortunately, such questions really only require basic knowledge of basic principles in these areas of mathematics, followed by an immense aptitude that you could only really develop by training for contest-style problems.

If I was to recommend a "priority list" for prepping for the Putnam, here's how I would rank book topics:

(1) Contest-style math problems (old Putnam tests and solutions, Art of Problem Solving, anything written by Titu Andreescu, etc.

(2) Set theory

(3) Number theory

(4) Analysis (calculus, real analysis, complex analysis)

(5) Combinatorics

(6) basic topics in abstract algebra

(7) basic topics in topology

Getting familiar with these topics should help you score well. It's a tall order, though. Most mathematics professors wouldn't be able to score well on the Putnam; it's not because their ability to teach and research is bad at all, but because contest-style problems really take training and develop to be tackled on site.

Good luck!

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