List of Math competitions I encourage students to try:
MOEMS (Elementary School)
Purple Comet MS
Andover Math Open (Hard) (sometimes local)
Lexington Math Contest (Middle School) (sometimes local)
AMC 10/12, AIME,
ARML (need to find local coach for tryouts)
Purple Comet HS
USAMTS (proof based)
College contests if you are able: CMIMC, HMMT, Duke, PuMaC, etc.
Math Prize for Girls
I have also heard good things about:
Some AoPS Mock contests are good and very high quality. I like the CMC ones.
Competitions that no longer run, but have good problems:
It is also great to try to make your own local math team or math club or math program. I highly encourage those trying to build a local math community to continue to try hard! Community is what keeps these things going, and having no local community in (insert endeavor here) makes it harder to succeed in said area.
FTW (for fun)
Tournament of Towns
Putnam (college, not olympiad)
The above all have good practice problems for olympiads. There are also geometry-only olympiads on AoPS forums.
Russian local olymipads have some good problems, same for some Eastern European Local olympiads. Other computational contests may exist in the UK, Canada, Singapore, India, etc, that may be at a good level for those preparing for AMC’s too.
Summer programs that I think are good
MathPath (middle school)
AweseomMath (middle and high school)
ROSS (NT Focus)
PROMYS (NT Focus)
RSI (11th grade)
Most of the above are for high schoolers
For latecomers to the math competition scene (i.e. starting in mid high school or as adults):
I encourage you to try some of the middle school math competitions, like AMC 8 or Mathcounts. Some of them are pretty tricky! Beast Academy also have some neat puzzles.
Advent of Code