The Putnam Mathematical Competition is the most prestigious math contest for undergraduate students in the U.S. and Canada. The exam is designed to test creative thinking in addition to technical competence, and is held every year on the first Saturday of December. The problems range across the undergraduate mathematics curriculum, but most do not require specialized knowledge of mathematics beyond calculus.
The best way to learn to solve Putnam problems is to start trying to solve some. Although no two Putnam problems are alike, as you start to see more of them, you gradually build up a store of intuition and ideas to draw upon in solving them. Even if you don't succeed in getting very far on many problems, you will find that the mere process of grappling with them strengthens your ability. For this reason, we hold a problem seminar at OU each fall to give students the chance to attempt to solve Putnam problems.
Everybody is welcome at the seminar, whether they're thinking of participating in the competition or not. Working on difficult, but interesting, mathematical problems is a good way to improve your mathematical abilities and have fun at the same time. You'll not only be better at mathematics proper, you'll be able to work better in any field of science or engineering that uses mathematics!
Below is information about this year's seminar, and at this link you can find some of the problems considered in past years.