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Harvard doesn’t release those kind of statistics, so it’s really anyone’s guess, A quick search of google yields this person’s guess, but I cannot say it’s accurate.
And then there’s this CC thread: http://talk.collegeconfidential.com/harvard-university/1340006-usamo-and-college-admissions.html
With several hundred qualifying every year, usamo/usajmo is probably roughly on par with (and about as helpful as) Siemens or Intel semi-finalist status. AFAIK the only STEM award or program that comes anything close to guaranteeing a spot at Harvard (probably > 80% acceptance rate) is RSI.
@tdy123 You can’t really look at it as “how many people get it”. Intel and Siemens have roughly a 20% acceptance rate as a semifinalists, which is somewhat contingent on luck, while USAMO has only a 1% qualifying rate from the people who take the AMC. More people take the AMC, thus the USAMO is probably more selective.
@AnnieBot All a person has to do to join the applicant pool for USAMO is show up for the qualifying test. No barrier to entry = large field = appears highly selective. Intel and Siemens have, comparatively, high barriers to entry: the students actually have to conduct novel independent research and write a research paper. High barrier to entry = small field = appears less selective. I suspect Harvard is aware of this.