Most of these require a registration fee, and most do not accept individuals to compete. Organize teams through your school or homeschool group.
Mathematical Olympiads for Elementary and Middle Schools: Elementary (grades 4–6) and middle school (grades 6–8) teams. Five monthly contests during the school year, no travel required. Students compete individually in their own schools or homeschool groups—with certificates for everyone and a top-scorer trophy—and then scores are compared nationally for additional awards. Low stress and lots of fun for beginning mathletes.
MathCounts: Middle school teams (grades 6–8). Students compete individually at the school level. Each school or homeschool group may send a team of four students to a regional competition, with the top teams progressing to state and national contests (travel required). Study resources available through the MathCounts website and at these links:
MathCounts Drills by Elias Saab: Tough online practice problems for MathCounts preparation, or simply to see if you can handle the challenge.
MathCounts Toolbox: This is a nine-page summary of the basic facts of elementary and middle school math. Go through each page, checking off all the things you know. Then try to learn at least one new math fact per week.
The MathCounts Bible According to Mr. Diaz: “What you must memorize, without excuses and for the rest of your lives (not just for MathCounts).”
American Mathematics Competitions: Middle school through high school teams. Students compete individually at their own schools, then scores are compared nationally. Awards given at school and national levels. No travel required. Past tests available through the website for study and practice.
Mandlebrot Competition: High school teams. Students compete individually and in teams of four. Ribbons given to top four scorers at the school level, additional awards at the national level. No travel required.
USA Mathematical Talent Search: High school, or advanced middle school individuals. Free. Students must solve challenging problems and write well-justified solutions. Four rounds per year, five problems per round, with one month to work each set. Past problem sets available through the website for study and practice.
Feature photo above by Fractal Ken via Flickr (CC BY 2.0). For more resource suggestions, check out my Math with Living Books pages.