The 2023–2024 USAMTS is over. See you in September!
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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.

Write Your Solutions

Unlike most other contests, you’ll have over a month to solve the problems and write up your solutions. You’re also allowed to consult certain reference material.

Example student solution: Let T be the set of squarefree integers less than 500. Let's calculate |T|. We can start with 500 and subtract the floor(500/2^2) multiples of 2^2, the floor(500/3^2) multiples of 3^2, and so on, for every prime: 500 - floor(500/2^2) - floor(500/3^2) - ... - floor(500/17^2) = 282. Thus |T| = 282. Now, we claim that T is an optimal choice for S, so our answer is 282. The rest of the proof is omitted for brevity.

Get Feedback

Our graders not only give you a score, but also personalized written feedback to help you develop your mathematical and writing skills.

Sample feedback: Score 3/5. Comments: Close! You are right that T is the optimal choice of S, and your proof of such is correct. However, you made a mistake in calculating |T|, which is actually 306. The issue is that you double counted numbers which had more than one squared prime factor, such as 36 = 2^2 * 3^3. To rigorously count the union of overlapping sets, you would need to use the Principle of Inclusion and Exclusion (PIE).

Win Prizes

At the end of the year, the top 45% of students will receive prizes, including math books and T-shirts. Prizes may include Wolfram software subscriptions as well. The USAMTS is also one way to qualify for the American Invitational Mathematics Examination (AIME).

Some sample prize books: Euclidean Geometry in Mathematical Olympiads by Evan Chen, 99 Variations on a Proof by Philip Ording, Introduction to Number Theory by C. J. Bradley, and Humble Pi by Matt Parker.

Interested? Join our mailing list to receive updates about the USAMTS, such as announcements that problems are available or that scores have been released. To compete, register today! Students can register any time during the year, even if the first round has already completed.