Below are some useful links related to mathematical problem solving, including a link to the William Lowell Putnam Mathematical Competition, a nationwide mathematics contest for undergraduates. All students in the course are expected to participate in this competition.
William Lowell Putnam Mathematical Competition
International Mathematics Olympiad Problems
American Mathematics Competitions Internet Home Page
USA/International/Asian Pacific Olympiad Archive
For those with some experience with proof-oriented math competitions, the first best step towards training for the Putnam is trying some questions from prior exams, to get used to the somewhat different style of the questions.
The best sources for past exam problems, with solutions and additional discussion, are the three Putnam books published by the Mathematical Association of America.
Gleason, Andrew M., R. E. Greenwood, and L. M. Kelly, eds. The William Lowell Putnam Mathematical Competition Problems and Solutions 1938-1964. Washington, DC: Mathematical Association of America, 1980. ISBN: 9780883854280.
Alexanderson, Gerald L., Leonard F. Klosinski, and Loren C. Larson, eds. The William Lowell Putnam Mathematical Competition Problems and Solutions 1965-1984. Washington, DC: The Mathematical Association of America, 2004. ISBN: 9780883854631.
Kedlaya, Kiran Sridhara, Bjorn Poonen, and Ravi Vakil. The William Lowell Putnam Mathematical Competition 1985-2000 Problems, Solutions, and Commentary. MAA problem books series. Washington, DC: Mathematical Association of America, 2002. ISBN: 9780883858073.
All of the problems are available (without solutions) at John Scholes's site.
Problems and solutions from the last 10-odd years, plus results of the competition, are available at the AMC Putnam archive.
Richard Stanley's Putnam Problems: A list of problems Prof. Stanley submitted for inclusion on the Putnam Exam when he was a member of the three-person committee to write the 1984-86 exams (PDF)