The great teachers of math to kids in the US are of course the Ross Program folks. (When last I checked in I thought Boston University's Promys was actually livelier than the original Ross Program at Ohio State.) They basically take very smart kids and help the rediscover much of abstract algebra and basic number theory.

Decades later, I probably couldn't state quadratic reciprocity any more, let alone prove it. But proving the Two Squares Theorem is still one of the best ways to clear my mind of less pleasant thoughts, should I need such a device. It's just a whole lot of fun, even if you go in assuming next to nothing. (I start by proving that the Gaussian Integers are a Euclidean Domain and that all Euclidean Domains are Principal Ideal Domains, and proceed from there. It doesn't take long.)