ARML (American Regions Math League) holds a yearly competition that I have coached teams for. My favorite thing about the contest is that there's a portion (called the "Power" contest) that requires teams to write proofs. Here is the official ARML page, and here is the homepage for the Chicago Teams
Art of Problem Solving is a resource for middle and high school students who are interested in math competitions - they have books, courses, active discussion forums, etc.
For 5th-12th grade students in Chicago who enjoy math and want to spend their Saturday afternoons doing it, there's the Payton Citywide Math Circle. Some of the organizers are fellow math team coaches and very inspiring teachers.