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A logical place to start for Americans is Mu Alpha Theta, the National Math Honor Society. From there, you can join many team-based competitions such as the Math League, Math-O-Vision, Moody's Mega Math Challenge, MATHCOUNTS, etc. For a more individual challenge, there is USA Mathematical Talent Search (USAMTS), which offers one month for solutions to be submitted, but offers the most extremely challenging problems.
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