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"I've actually heard that usamo is a huge boost at caltech and comes close to guaranteeing admission. Not sure if this is true though. One that's for sure is that the usamo qualifiers i know are math geniuses, as in acing calc bc in 8th or 9th grade. Qualifying for usamo shows you can really think logically and outside the box, better than almost anyone. Isn't this invaluable to engineers?"
Not true. Caltech does reject USAMO qualifiers. Like simfish said, a lot of these math contests have to do with growing up in a culture that emphasizes mathematical pursuits. In a weird way, qualifying for the USAMO is a lot like doing well on the SAT.
I know this Chinese kid who was both a USAMO and US Physics Semis qualifier, but was rejected from both Caltech and MIT. His SAT score was like a 2000, and it was clear that he was only good at contest math because of the intensive training he received in China.
I don't know if you're speaking of "geniuses" in absolute terms, but I can definitely tell you that to people who are a part of some problem solving culture, USAMO qualifiers are a dime a dozen. Also, anyone who's practised extensively for the AMC series contests will know that hard work is oftentimes, sufficient enough for success in USAMO qualification. I don't know what you mean by this "thinking logically and outside the box" stuff, but I really don't think USAMO qualification is indicative of "genius" at all. A good year of solid prep, often does the trick.
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