Math competitions for hobbyists?

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Are there any math competitions for hobbyist / amateur mathematicians? Something like the Putnam or the International Mathematical Olympiad, but open to regular people who are not full-time students?

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You can "unofficially" (well, nobody really keeps track) participate in (mostly short answer, but some proof-based) student-run contests like the NIMO and OMO (see http://internetolympiad.org/), or the (proof-based) Olympiad-style ELMO (see this year's AoPS-version of the contest). Of course, AoPS also has a good repository of past contest problems; in particular, for the aforementioned NIMO, OMO, and ELMO (including ELMO Shortlist), you can find corresponding archives in the USA Contests subsection.

(Disclaimer: I was heavily involved with the OMO and ELMO in high school.)

In any case, AoPS is probably the best place to look for such contests (at least those based in the US). And as Will Jagy's comment suggests, high school students are probably much more willing to spend time and energy writing these contests than others.