The AMC (American Mathematics Contest) is a highly prestigious test that is part of a sequence that culminates in the International Math Olympiads. Students who do really well are invited to the next levele - the AIME.  The questions are very good ones (probably because it is sponsored by prestigious societies of real mathematicians) that progress from very easy to very hard.

The Math Olympiads are a nice, low-key, ongoing learning experience. The monthly tests are short and then right afterwards we discuss the problems, which help develop creative thinking skills. There are lots of small awards, which are presented at the SBFS end-of-year family celebration ceremony.

The CA Math League is similar in format - short, good, creative problems - but with fewer prizes. Although the high school version is monthly, there are annual versions for younger kids available.

The Bay Area Mathematcal Olympiads (BAMO) consist of 5 engaging problems with no time pressure. 3 of the problems are in common, preceded by 2 easier ones for BAMO-8, or followed by 2 harder ones for BAMO-12. The award ceremony is about 2 weeks later in the Bay Area.

The RCML (Rocket City Math League) is similarly structured, but the levels are based on subject matter (Pre-Algebra, Algebra I, Geometry and Algebra II) instead of grade, the awards are bigger – and it's run by students! We do not participate in it every year.

The ARML (American Regional Mathematics League) Competition takes place over two days on a college campus, with the contestants staying in dorms. The competition includes fun "mathletic" events like relay races. Teams consist of 15 students and can come from any size geographical area, including states. We join together with groups from the Los Angeles area to form multiple Southern California teams. Practices are held once or twice a month on weekend afternoons. The level of this competition is very challenging. It is the only one where students from different schools and homeschoolers all work together.

The Caltech Math Meet, formerly known as the Caltech-Harvey Mudd Math Competition, is a challenging all-day event on the Caltech campus, preceded by an online Power Round.

MATHCOUNTS is the premier national middle school math competition, involving a partnership of industry and academia (with the national winners going to the White House!). The organization provides lots of materials for coaching.

The MathCounts Club Challenge contests are similar to the Olympiad ones described above.

The Math SuperBowl is an exciting local meet that gets bigger every year. It is sponsored by Montecito Union and a few other schools. Teams practice throughout the year from previous years' tests.

Math Day at the Beach is a long drive - but I love the name and it's fun as well as very challenging.

The MC⁴ (Math Council California Central Coast) competition was unfortunately terminated over a decade ago.

The PiMC (Pi Math Contest) has three levels.   They are  like low-key AMC contests, with the youngest one being like an AMC 5.  It is organized by AlphaStar Academy and the awards include generous coupons for their programs.

The Westmont Math Field Day, formerly known as the Westmont Math Competition, is an almost-all-day event, including gameshow-style contests, talks and dinner.