> By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. No, those are table stakes. To

> By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. No, those are table stakes. To reach that level you have to know the theory and be good at the game. > Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true. By that standard pretty much 100% of mathematical education is silly tricks unless you are literally getting a PhD in pure math. Calculus without real analysis? Silly trick! Statistics without measure theory? Silly trick! Learning how to code without an in-depth understanding of computability theory? Silly trick! Applying Burnside's lemma in a competition setting requires at a minimum a fairly sophisticated understanding of what is a group action and an orbit, which is definitely "real math". The broader point is that competition math isn't some kind of parallel universe where you learn "fake math" that doesn't exist in the real world. You learn the real thing -- perhaps less rigorously than in a university course, but the real thing nonetheless.