I am interested in applications of somewhat "advanced machinery" (with respect to the usual machinery involved in these cases, which is usually elementary) to olympiad or (high school-level) contest problems in mathematics which yield simple, insight-provoking solutions.
This may be something as simple as basic group theory or linear algebra, for instance. I shall post an example of what I'm looking for as an answer.
Please don't post joke-ish things like the "Fermat's Last Theorem is too weak to prove that $\sqrt 2$ is irrational" thing, please.
Perhaps we could make this CW and have separate answers, each on applications of a specific area of mathematics?