What extracurriculars help with a mathematics major application?
The strongest math extracurriculars show engagement with mathematics beyond coursework: a competition pathway from AMC through AIME, attendance at a selective summer program like PROMYS or Mathcamp, and ideally something you created, whether a math circle, a tutoring program, or original research. Admissions readers at top math departments want to see that you do math for its own sake, not only to get grades.
How important is USAMO for top math programs?
A USAMO qualification is one of the strongest single signals for elite math programs like MIT, Harvard, Princeton, and Caltech. It is not strictly required, but the overlap between USAMO qualifiers and top-admit math students is very high. If you are within range of qualifying, treating the AMC-to-AIME-to-USAMO path as a multi-year goal is worth the investment.
What if I am good at math but not at competitions?
Research and founding activities carry comparable weight. A student who attended PROMYS and wrote a strong original problem set or did genuine number-theory research with a professor is just as compelling to admissions readers as a solid AIME performer. Competitions measure speed and pattern recognition; research and teaching measure a different and equally valued kind of mathematical maturity.