Note: this is now re-posted at mathoverflow here

I occasionally teach high-school students to prepare them for various math olympiads (with the final goal being the International Math Olympiad, IMO. It's my impression that over time the "curriculum" of the IMO and math olympiads more broadly (admittedly a vaguely defined concept) has been gradually shifting towards some of the more advanced and technical topics traditionally associated with university-level mathematics education. To name a few examples, we've already seen applications of linear algebra in combinatorics, the combinatorial Nullstellensatz, and the probabilistic method.

However, in mathematics we often value generality above accessibility (and for good reasons), which leads to some very beautiful ideas being inaccessible to the olympiad community until the person with the right intuition, examples and exposition comes along at the right time to transmit those ideas.

Thus, I've been wondering if we can compile a list of such ideas (together with illuminating accessible examples and expositions) which have recently entered the olympiad culture, or which are now becoming ripe for entering it.

Just one example that comes to mind: Fourier analysis on cyclic groups is a very accessible tool which has some rather deep consequences, such as Roth's theorem on arithmetic progressions (a self-contained proof can be found in these notes from a course taught by Tim Gowers). Thus, I wonder whether there might be nice examples of the use of Fourier-analytic ideas of this flavor in math competitions.


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