The Putnam is a challenging opportunity for you to test your mathematical mettle. In the 2014 competition, about 4300 individuals participated, representing over 550 colleges and universities across Canada and the U.S. The Stanford team placed seventh, and many individuals did very well. The measure we look to is the number of competitors on the list of top performers sent out with the results.
The competition emphasizes ingenuity rather than knowledge, so freshmen are not at much of a disadvantage compared to seniors. Interest in or experience with problem solving is a plus. Not just math majors have done well; many recent winners have come from throughout the mathematical sciences, including physics, computer science, and engineering.
Completely solving even one of the twelve problems is a significant achievement, and in almost all years would place you well above the median. (Keep in mind that the particpants are self-selected from among the best in the continent.)
You can just take the Putnam without thinking about it beforehand. But you are also welcome to come to the Pólya Problem-Solving Seminar held on Monday evenings over dinner. And you can also check out these resouces. (Here is the official Putnam website.)
If you are (even potentially) interested in trying the Putnam, please let us know (by emailing either of us). We maintain an email list (not the coursework list associated to Math 193) for those interested, and we will announce the location, etc. there.
Want more? Additional suggestions for others? Please let us know!