How important is a good AIME score for college admissions?

We’ve had a few really exceptional math kids who have had very high scores on these. They have been accepted to MIT, Stanford, etc. I suspect that top standings were further validation of their abilities. I believe, however, that they had other things in their application that pointed to this as well.

So yes, my guess is that a top score is a way to differentiate yourself from the crowd. And that a kid who gets a top score may already have demonstrated that s/he is not one of the crowd in some other ways.