Elegant applications of advanced techniques to "olympiad" problems

Question

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.

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Perhaps we could make this CW and have separate answers, each on applications of a specific area of mathematics?

Answer 1

An example would be Tim Gowers' beautiful proof of the following problem:

Let $n$ be an even integer. How many subsets of the set $\{1,2,\dots,n\}$ can you pick if they all have to have odd size but the intersection of any two of them has to have even size?

using basic linear algebra over finite fields.

Concisely, he considers the characteristic vectors of these subsets and shows that they must be linearly independent over $\Bbb F_2$ for the condition to hold, so the maximum number of such subsets is just $n$.

Takeaway idea: the dot product of the characteristic vectors (over $\Bbb Z$, of course) is equal to the size of the intersection.