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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There are lots of good books.
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.
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)
(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.
Putnam does not require any knowledge of analysis/algebra/topology, just do past exams like suggested above by user6312.