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Is synthetic geometry still relevant in mathematics? I always saw Olympiad geometry as an odd field because while every other Olympiad topic would extend to larger math, geometry seems especially useless. I get that analytic geometry is useful in college but that is heavily discouraged in Olympiad geometry. So what does olympiad geo extend to?

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    There is still work done in Euclidean geometry. My understanding however is that it is not a good strategy for getting a job as it is not frequented. – 2017-02-22
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    Any geometry problem can be mapped to an analytic geometry solution. But it is usually more work to use the analytic solution, even for trivial problems. Non analytic geometry (including Olympiad geometry) it's just a shortcut to get the desired result. The more shortcuts you know (Olympiad geometry), the more chances you have to get your proof with fewer calculations – 2017-02-22
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    Euclidean geometry is "consistent and complete" which may make it boring. Or, give the sense that all of the interesting things have been discovered already. As far as a pedagogical tool though, it is easy to visualize and provides an introduction to proof. And, it holds an important historical place as for most of the last 2000 year Mathematics WAS Geometry. – 2017-02-22

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Here is a list of open problems in Euclidean geometry taken from Mathoverflow. Thus, the sentiment that "all of the interesting things (in Euclidean geometry) have been discovered already" is just false.

Edit 1: At the same time, most MO problems have no connection to the modern research.

Edit 2. Here is one (admittedly atypical) example. I was told by a person present at the conversation, that the paper W.P.Thurston, "Shapes of polyhedra and triangulations of the sphere" grew out of a conversation about a Brazilian MO problem that Thurston had with a Brazilian grad student in Princeton around 1984 during a lunch break. (Sadly, I do not know the name of the student and the exact MO problem.) According to Google Scholar, Thurston's paper currently has 233 citations. Of course, one has to be William Thurston to accomplish such a feat.