The William Lowell Putnam Mathematical Competition is a nationwide mathematics competition which takes place in December. The description of the Putnam competition, taken from the web page referenced above, is reproduced below.

The examination will be constructed to test originality as well as technical competence. It is expected that the contestant will be familiar with the formal theories embodied in undergraduate mathematics. It is assumed that such training, designed for mathematics and physical science majors, will include somewhat more sophisticated mathematical concepts than is the case in minimal courses. Thus the differential equations course is presumed to include some references to qualitative existence theorems and subtleties beyond the routine solution devices. Questions will be included that cut across the bounds of various disciplines, and self-contained questions that do not fit into any of the usual categories may be included. It will be assumed that the contestant has acquired a familiarity with the body of mathematical lore commonly discussed in mathematics clubs or in courses with such titles as "survey of the foundations of mathematics." It is also expected that the self-contained questions involving elementary concepts from group theory, set theory, graph theory, lattice theory, number theory, and cardinal arithmetic will not be entirely foreign to the contestant's experience.

Below are two links to archives of problems (and solutions) to problems from past competitions.
problem archive 1
problem archive 2

Results from the sixty-fifth Putnam Mathematical Competition

Congratulations to Peter Thayer who performed well in the sixty-eighth William Lowell Putnam Mathematical Competition. Peter's ranking was 431.5 (out of 3753 participants with 1 denoting the highest rank).