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<p>Out of the 250-300 who take the USAMO, the majority make zero, or close to it, because they do not actually know how to solve and write up USAMO type problems. It is very different from the type of problem solving they have to do to qualify in the first place. The kids who do know how have taught themselves, taken useful college level courses, or learned at special camps. Artofproblemsolving should make this sort of mathematical thinking much more accesible in the future. They offer online classes that are an introduction to USAMO type problem solving and proof-writing. Some of the other contests mentioned, like USAMTS, are much easier than USAMO and are also a way for a student to get their feet wet.</p>
<p>The problem solving students do in preparation for competitions is much superior to the way math is usually taught. I don’t think kids need to actually compete to get the benefit. Many math kids prefer not to compete, but would enjoy working on competition-style problems.</p>