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 success:

1. First …

I was in Boston over this weekend for the 2014 Harvard-MIT Math Tournament. Before the contest on Friday, I sat in a few of the undergraduate math classes. They were pretty nice; I was actually able to learn some higher math that just by sitting in, despite the fact that I didn’t have the necessary background. I also got the feeling that the lectures moved somewhat slowly, which is probably how I managed to follow what was happening.

Anyways, I promised a sampler, so attached (at the end) are the notes I took during the classes. As I mentioned, I figured out what was happening in the first two lectures but not the third (so the notes for that one might be total gibberish). Hopefully they’re somewhat interesting though :D

HMMT 2014 Sampler