I always wondered whether I could generate olympiad geometry problems by simply drawing lines and circles at random until three lines looked concurrent, four points looked concyclic, et cetera. From extensive experience you certainly get the feeling that this ought to be the case — there are tons and tons of problems out there but most of them have relatively simple statements, not involving more than a handful of points. Often I think, “I bet I could have stumbled upon this result just by drawing things at random”.

So one night, I decided to join the tangency point of A-mixtilinear circle with the orthocenter of a triangle ABC. You can guess about how well that went. Nothing came up after two hours of messing around randomly.

Surprisingly, though, I found almost by accident that the following modification has had significant succes:

- First, start with a known configuration: for example, the the incircle and A-excircle touch BC at D, E such that BD = CE.
- Do some random manipulations to obtain an equivalent result. For this step, I like to use harmonic bundles, but I suspect other things work too. At the end of this step you should have something you might call a “proxy problem” — while you could take this and propose it to an olympiad, it’s probably somewhat contrived and uninteresting at the moment. (Your mileage may vary! I did get a nice result out of this once.)
- Starting with the proxy problem, start adding in new lines and circles and points until you find a conjecture.

In other words, it seems like “randomly wander” doesn’t work so well, but “randomly wander starting from somewhere that already has structure” works great. I’ve tried this four times and each time has gotten me a new problem.

I have to wonder what makes this work. I think it has something to do with the fact that olympiad geometry has a different structure to it than the other olympiad subjects. For example, given an arbitrary geometry problem and two contestants, it’s much more likely that they will come up with different solutions than in any other subject. This seems to suggest that, if one takes a true result and examines the diagram, there is likely other structure present within.

That’s my best guess for why this works, anyways. Obviously just heuristics, but as long as I keep getting more problems…

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## Published by Evan Chen (陳誼廷)

I am a math olympiad coach and a PhD student at MIT.
View all posts by Evan Chen (陳誼廷)