>>42
You're eligible for the Putnam as long as you haven't gotten a bachelor's yet. The knowledge base required is restricted to basic undergraduate stuff: Calculus, rational number theory, linear algebra, and combinatorics. Maybe graph theory, complex analysis or group theory here and there. The purpose of the test, though, isn't to test your subject knowledge, but to test cleverness, creativity, speed, and general problem solving ability. On the other hand, if you're not *absolutely* fluent with these subjects to the point that they are second nature, to the point that you can solve the relevant questions on those TRIPOS tests almost without thinking, you'll be seriously handicapped during the actual test. Many people who wind up with PhD's from top universities (including one from MIT that I know personally) never score higher than 10 or 20 out of 120.
If you haven't already found it, here's the official problem archive:
http://www.unl.edu/amc/a-activities/a7-problems/putnamindex.shtml
Problem #1 on either half is typically a giveaway, and #6 is typically next to impossible. The ones in between vary randomly in difficulty.