North Suburban Math League
http://nsml.org/
Team event - up to 41 students. 5 problems per grade level, plus oral contest
5 weeknight meets per year
ICTM State Math Contest
http://www.ictm.org/links/contest.html
Team or individuals. Several different events at all grade levels
2 Saturdays - regional in late February, State in late April or early May
American Mathematics Competitions (AMC)
https://maa.org/math-competitions
Individuals - 25 question multiple-choice exam. 2 levels
70 minute exam. Later rounds require more time for those who qualify.
Chicago ARML Teams
http://chicagoarml.org/
Individuals tryout for teams. Teams compete in 4 different events. (For more advanced students)
Tryouts - 1 weeknight. Practices - 4 weeknights. Contest - one weekend
ARML Power and ARML Local
Info at http://www.arml.com/.
Team events. Power involves written solutions and proofs to themed question (For more advanced students)
Power - 2 1-hour contests during year. Local - one Saturday contest.
Mandelbrot Contests
Historical. The competition was permanently discontinued in 2019.
Illinois Math League
http://www.mathleague.com/
Individual contests. 6 problems per test.
6 short tests throughout the year.
American Scholastic Math Association
https://www.asan.com/annual-math-contest
Individual contests. Similar to Illinois Math League.
6 short tests throughout the year.
Purple Comet online math contest
http://purplecomet.org/
Teams of 4 students solve 25 problems.
One 90-minute test in the spring
Mathematical Contest in Modeling
https://www.comap.com/contests/mcm-icm
Teams of 4 students create mathematical model for real-world problems. (For more advanced students.)
USA Mathematical Talent Search
http://www.usamts.org/
Individual problem sets (For more advanced students)
For students to register and do on their own
MathWorks Math Modeling Challenge
https://m3challenge.siam.org/
A modeling contest for high school juniors and seniors.
National Assessments and Testing
http://www.natassessment.com/
(Apparently defunct.) Various team and individual contests.
Format: 4 hour-long contests, 1 week-long collaborative effort.