Does there exist an abelian group where the product of the orders of the members is 2^2009?
I vaguely remembered that an abelian group is one that is commutative (which is correct), but also that I didn’t remember anything else so I would never be able to do the problem (which is also correct). I wrote “No” and moved on — I ended up being right, and it didn’t actually to say to prove it… The next problem I worked on was a nasty calculus problem. I spent most of my time in the first half antidifferentiating things like 1/(x^2+1) and tan(x).
There were two other approachable problems in the first half, but I was annoyed by the amount of formal math that you needed to do the others. I suppose, though, that given the kind of competition that the Putnam is, and the fact that I haven’t taken that much math in college, I shouldn’t be surprised that I was unequipped to do some of the problems.
The second half was a lot better. The first 5 problems were approachable; the 4th talked about the dimension of a certain vector space but that was fairly intuitive. B5, though, was a problem about the limit behavior of a function defined through a differential equation. The intuition was easy enough to know that it was going to grow without bound, but since I don’t know any formal analysis I wasn’t able to relate local properties of the function to global properties, even though the link is “obvious”. Again, I was lacking the machinery to make the proof fully rigorous.
I think this says more about me and my math training (or lack thereof) than about how the Putnam should or should not be.
Anyway, now begins the long wait until results come out in spring. I’m probably going to forget everything about this Putnam by then, but hopefully I’ve reversed my declining streak that I’ve been going through the past few years.