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Sample problem: Let S be a subset of {1, 2, ..., 500} such that no two distinct elements of S have a product that is a perfect square. Find, with proof, the maximum possible number of elements in S.
Math Online

USA Mathematical Talent Search (USAMTS)

by USA Mathematical Talent Search

USAMTS is a free, proof-based online math competition for US middle and high school students. Students have a month or more to work on problems. Registration is open year-round.

Location
online
Cost
Free

About

The USA Mathematical Talent Search (USAMTS) is a free, proof-based mathematics competition for all United States middle and high school students. Unlike many competitions, USAMTS allows students a month or more to work on problems, which require carefully written justifications. Students can register any time during the year, even after initial rounds have begun. The competition consists of multiple rounds throughout the academic year, with a new cycle typically starting in the fall. Participants may use resources like books and the internet for research but cannot receive 'live' help or discuss problems with others before the submission deadline. Solutions are submitted online, and participants receive personalized written feedback. USAMTS is also a pathway to qualify for the American Invitational Mathematics Examination (AIME).

Who's eligible

All United States middle and high school students.

How to enroll

Online submission of carefully written justifications.

Ready to apply?

Information current as of Jun 11, 2026.