well I guess it's only Day 1.

Darn, I envy you guys in the US for actually having decent problem setters. Also, you can discuss the problems in about a day. (Nearly two months after APMO we're still waiting.) I actually resisted looking at the solutions until I briefly thought through the problems at home this time. I'm not competing in olympiad math this year (I'm studying abstract algebra (well, duh) and writing a math paper of sorts), and probably not in the future again either, but I found these problems really interesting. Even the geo.

1: Haha, geo. I struggled and after a while came up with a ridiculously horrible solution inverting about and using a dozen Law of Sines and powers of points and inversion-induced similarities and stuff. And I thought I had found a non-bashy solution. *facepalm* I'm going to give myself the benefit of the doubt since I was too lazy to get out my compass.
2: Coming up with the bijection was quick, although I changed my mind about whether it actually worked four times or so. Blargh.
3: Overlooked a case --- nope, there are more effectless flipsets than all of two orientations --- and read too much of the solutions. Meh. I think this is a bit messy to solve.