<p>Thanks, that is very clarifying. I guess the one sticky point in all this is that not everyone who takes some classes past calculus and basic college physics has the same interest, aptitude, etc in those things. I think unfortunately, this is a bit tough to gauge, especially if the candidate is not mature enough to express his or her interest in a distinguishing fashion. </p>
<p>I am interested in all this partially because I did not even know what the USAMO was until well into college, and I know a fair number of enthusiasts who are somewhat the same.</p>
<p>One interesting thing is there are tenured professors at the same (super top) places both with IMO gold medals and no track record whatsoever. Clearly, they have abilities beyond what many others with no IMO/etc track record do, so I imagine letters and other things are pretty crucial to getting the full picture.</p>