Putnam Advice for Non-Olympiad Students

Be prepared, it's a long one.

So you're going to take the Putnam for the first time. Big deal, right? You are a straight-A maths student after all, and got perfect or near-perfect on SAT reasoning and Math subject tests. You may have heard about the legendary difficulty of the exam, but what could be so hard about it, right?  It's just math -- and any problem can be solved with a clear head and a good night's rest, right?

The truth is, if it's your first time trying your hand at Putnam-style problems, you are probably going to be in for a world of hurt. The top scorers on the exam typically had exposure to these kinds of problems in high school through contests like USAMO or IMO.

***DIGRESSION***

• Q. I'm a high school student. How do I participate in USAMO or IMO?

• A. Typically top scorers in USAMO participate in IMO.  The caveat is that, in order to participate in USAMO, you need an invitation, and invitations are not simply given away. You first need to do well in both the AMC 12 and the AIME. So if you're still in high school, read on -- but pay special attention to the resources available for AMC 10/12 and AIME preparation. Though be warned that I am pretty unqualified to give advice in this area.

***END DIGRESSION***

I've seen contests like the Putnam do a serious number on the confidence of some very intelligent people, and I think that it's really unfortunate that it tends to happen that way. My experience is of course biased toward what I saw at my undergraduate institution, but I suspect that, at lots of places it goes something like this:

• Gifted Freshman is invited by math Professor to do Putnam.
• Freshman shows up the Saturday morning of the exam to do Putnam. Lots of other freshmen present, also a few masochistic upperclassmen vets.
• Freshman takes first half of exam. HOLY CRAP THESE PROBLEMS ARE HARD.
• Feeling drained, Freshman unenthusiastically returns for the second half of exam after lunch.
• HOLY CRAP THESE PROBLEMS ARE STILL HARD.
• And yet, Freshman tried to write something for some of the problems. In March, Freshman hears back from Professor. He got a 1. Out of 120. Spirits crushed, he vows never to return to that cursed exam. And he forever feels inferior to his friends that got scores in the 10's or 20's.

*******

The truth is, the Putnam is like anything else: practice makes perfect. (See this post on contest math.) Your friends and their fancy non-single-digit scores probably did some prep for math contests in either college or high school, or maybe they did engineering competitions like TEAM+S in high school, or maybe they did programming competitions. Whatever the case, they probably had some experience thinking analytically. And you can, too. Below is a list of resources I wish I had known about as a freshman in college, and that I wish I had spent more time on once I did become aware of them.

Beginner Resources

Way back in high school, I had very little perspective on how little math I knew. If any of number theory, analysis, linear algebra, or abstract algebra (NOT middle / high school algebra / precalculus) sound unfamiliar to you, you definitely want to take a look at some of the following and maybe make some purchases from Amazon:

1. Proofs. For my "intro to proofs" class, we used A Transition to Advanced Mathematics by Smith et al. I considered it a generally good read, though the price is quite ridiculous. I recommend checking out other books on Amazon before sinking any money into that thing.
2. Algebra. A good modern / abstract algebra book should read like a bedtime story, and Charles Pinter's book does exactly that. Another good introductory text comes from Durbin.
3. Number theory. This book is a nice, accessible, cheap little text from Underwood Dudley. I also recommend searching Amazon for more advanced texts.
4. Analysis. I'm pretty weak here, but I had a great experience with Bilodeau's introductory text. It's a little pricey, so you may try one of the Dover books instead.
5. Linear Algebra. I absolutely adore Carl Meyer's book.  One caveat is that the book is fairly comprehensive, and it covers much more linear algebra than is needed for competitions. Still, you should read it for the pure joy!
6. Differential equations. I am not qualified to make recommendations here; still, you can't go wrong with Amazon.

If you're still in high school and want to prepare for AMC and AIME competitions rather than delve into annoyingly long books, I've heard great things about AOPS volumes 1 and 2.  Also check out some of the AMC and AIME archives (problems + solutions!).

Intermediate / Advanced Resources

I'm not going to flatter myself and put myself in the "advanced" category; still, I benefited greatly from the following:

1. Art Of Problem Solving (AOPS) articles.  By far the most comprehensive resource for competitive mathematics preparation, IF you can navigate the site. This wiki page has lots of links to various resources, but lots of them are broken. If you already have some experience with Olympiad-style problems, check out the articles on this page. Thomas Mildorf's article on inequalities is particularly excellent.
2. Putnam and Beyond, a great (free!) book with lots of practice for Olympiad-style problems.

And, again, I've heard great things about the Olympiad prep books on AOPS.

So hopefully you have some direction for Putnam preparation, and for general contest math preparation.  Don't get discouraged by poor performances -- it's all about practice, practice, practice!  Being able to solve a couple problems on the Putnam is immensely satisfying, and I've heard it doesn't look too bad on CS / math grad school apps either. :)  (That being said, research should be your primary concern if grad school is your goal, particularly for CS.)  Best of luck!

-Stephen

A Quick Life Update

Just  a very quick post... This summer I'll be working at Palantir, and after that I am going to Stanford for a funded MS in computer science. So the PhD will have to wait if it ends up happening, but for now I am very happy with things. :)