<p>I would like to give a unique insight in this debate, as I have never qualified for USAMO and never have played a real sport in high school myself (unless you consider ping-pong, badminton, and intramural tennis a sport). I have friends who have done both, however.
I would give the edge to USAMO (possibly biased, because I’m president of the math team and have helped a friend of mine make USAMO). The questions on the AIME are <em>gasp</em> hard, and all I could muster was a 3 (I made so many careless errors), with a decent amount of practice. It’s hard to solve AIME problems, and requires an innate ability to think creatively and to go outside the box to do well. I admit, I’m not a theoretical math person because I am much more practical math and have a better ability to lead and organize projects (which is why I’m going into finance and not math).
It requires a bit more training, albeit very different from BOTH academics and sports, and I regard those that qualified to a very high degree.
Now, I know that varsity sports are not nearly as hard to get in as USAMO (100+ of my friends are or were on varsity teams), but that is a bit unfair to the varsity person. I will then, compare D1 to USAMO.
There is a kid at my school who swims in DI/DII, and it takes him 5 hours of practice a day year-round. His grades are not nearly as high as mine because he doesn’t come home until 9 (and tends to slack off in school). I respect him for his hard work and don’t make fun of his un-Asianiness too much, but the amount of training that takes does not rival the amount of training a trained ballerina does (I have a friend who both was on varsity cheerleading and danced for a private ballet company), or even as much as USAMO qualifiers put in. I have more respect for a highly trained ballerina with mediocre grades that a DI varsity with the same grades, even though DI varsity, as many stated above, is good profit for the school (the Ivy League is a sports league after all, not an academic league). This is the reason why schools are more lenient to athletes.
And to break it to you, USAMO qualifiers do NOT always get 2300+ and 4.0 GPA’s. This is because many of them are relatively <em>gasp</em> weaker in English and History. It is much more common for USAMO qualifiers to get ~2200+ and a 3.7-3.9 because of this “weakness” (albeit I only know about 10 or so USAMO qualifiers during my 2.5 years, so I could very well be wrong). I’m only speaking from experience, so don’t dinge me on this.
Although it doesn’t matter for me anymore. I’m going to college. :P</p>