<p>I thought I’d clarify about the USPhO semi-finals. The number who qualify is a little more than the number who qualify for the USAMO. I think the USPhO have about 320 and the USAMO around 220-250 (though maybe someone could get the exact numbers); so, I think making the USPhO is definitely a big accomplishment comparable to the USAMO, particularly to colleges that would like to have top physics students in their schools.</p>
<p>As to the comparison between AIME and getting an 800 on SAT I Math, there really is no comparison in my mind. My son easily scored an 800 on math in 8th grade on the SAT I but he did math competitions for the first time last year as a sophomore and only got a 3 on the AIME. I am sure he’ll do better this year after taking an AIME AoPS class, but the math topics covered by the AIME as far as I can tell aren’t part of a typical high school curriculum. I know next to nothing about math but I’m pretty sure things like number theory, combinatorics, and probability are not covered in a typical high school curriculum.</p>