So it turns out on three of the problems (A1, B2, and B3) I had a pretty good argument for each, and now I just have to wait until March to see if I got any points. The solutions are posted here. I would be interested in hearing from any of you, your ideas about the other solutions as I didn’t come close to any of the others.
A little bit of history on the Putnam: only three have ever achieved a perfect score on the Putnam exam. David Moews in 1987, it having been a highly difficult year for the test. Two in occurred in 1988, although I can’t pin down who they were. Although the top five, had been fellows at least once, often several times before. You can read up on a little bit of history here.
As for other mathematical competitions there is the International Mathematical Olympiad, which if you look at the problems these high-schoolers have to complete, you’ll be impressed. Then there is the Mathematical Contest in Modeling which a professor here in BYU-Idaho is very interested in registering some of us students for, although it has been hard to pass by the administration because the actual contest takes four days, and we pretty much have to eat, sleep and not shower at school. It’s heavily programming oriented and has had some very interesting problems to model in the past including “finding the sweet spot on a baseball bat” (2010), finding the location of serial criminals, among other things. Here are some other competitions.