The Purdue University Math Department publishes a math problem every week to encourage interest in undergraduate mathematics. The problems can be found here.
If you are an undergraduate and think you can solve this week's problem, write up a solution and mail it to:
Problem of the Week
Department of Mathematics, Purdue University
150 North University Street
West Lafayette, IN 47907-2067
A collection of math contests of varying difficulty for those interested in competition math.
The collection can be found here