The tippy-top schools are actually quite interested in math competitions too. The premier high school contest is USAMO. About 250-300 kids per year qualify; each year the top 12 scorers are named winners, and the next 12 get honorable mentions.
I took a look at four years of winners from 2013 through 2016. (Most 2017 and 2018 winners are either still in high school or just graduated, so it’s tough to find information.)
Here are the stats for the winners, 48 named.
Number of unique winners (some students repeat): 36 (34 boys, 2 girls).
Colleges attended
MIT – 17
Harvard – 11
CMU – 2 (both are named Knaster-McWilliams scholars and one is also a Goldwater scholar, so full rides+)
Princeton – 1
Ohio State – 1 (student from Michigan; OSU is also home campus for the Ross number theory program and research)
Unknown – 1 (common name and I gave up looking)
Still in High School: 3
As I basically knew, USAMO winners are essentially auto-admit at any school that they would want to attend. In googling around, I often came across statements like this: “Now 17, he’s going to Harvard, but all the other Ivy League schools accepted him too, plus Stanford, MIT, Carnegie Mellon, Chicago…” And he was only a winner in one year. Of course, it’s not the contest win that gets the kids in. The contest win is strong evidence of >+3.5SD sort of intelligence. There are very few kids that smart even at HYP, way less than 10% of the class. (That’s also why it’s trivial to predict that a high AIME score will, for instance, garner a very high SAT score, assuming even basic exposure to English - it doesn’t take too long to get fluent for those kinds of minds.)
I didn’t go through all the honorable mentions (48 of them named in those four years), but many of the names appear all over the recent Putnam lists, and seem to be at tippy top schools (MIT, Harvard, CMU, Princeton, and others). All USAMO qualifiers are in very high demand.
I also briefly chatted with an acquaintance who is very high up in the competition world (full professor with more than 20 years’ experience in this) who is also very interested in increasing access for girls in math (as all the schools are), and he said that simply qualifying USAMO is enough for girls to basically “go anywhere.”
Last, for people out there who listen to curmudgeons like Cathy O’Neill, who pooh-poohs contests as not being real math, a few of the 2013 winners are already pursuing research mathematics graduate programs at Princeton, Berkeley, MIT, etc. So, contests are not dead ends. You will often see bios like this if you dig around:
Hope that info helps some people out there! I have no idea why the kid up there applied to so many schools (he was also IMO Gold) - what a waste of time!! But then again, I doubt he spent much time on his essays, and as I wrote somewhere above I doubt that they were even read before admitting…