This is a little overdue, but here are my thoughts on this year's USAMO.

Too much number theory. But other than that, I had fun with the test.

#1: A good exercise for beginners to show they know how to write a clean proof. A little playing around quickly reveals what numbers work, but beginners will struggle to write a complete proof. Therefore, I think it's a fine #1. I'd say that this problem is a great argument for why it would be great if more students were allowed to take the USAMO, but I'll blog more about that later.

#2: Took me way longer than it should have, particularly considering that the means mod 19 insight is essentially equivalent to a problem I teach in the Olympiad Problem Solving class (Prove there are no solutions to .) Of course, we here at AoPS have to love this problem, not only because we taught it in the Olympiad Problem Solving class, but because one route to the factorization lies in seeing that we can use Simon's Favorite Factoring Trick. And we love saying 'Simon's Favorite Factoring Trick'.

#3: My kind of geometry problem. Leave it to Zuming to write a problem that looks like it should have a very simple 2-step angle-chasing solution but doesn't. I love that it can be solved with a series of observations using very basic tools - cyclic quadrilaterals, inscribed angles, parallel lines, reinterpreting the problem in another form. Collinearity problems are often the hardest in geometry. In the Olympiad Geometry class, we teach one approach to these is simply to reinterpret the problem as something else, usually a concurrency problem. Typically this is a good route because when trying to prove collinearity, it's often easy to mistakenly assume the points are collinear and come up with a bogus solution. (I, of course, did this several times in the course of solving this problem, as DPatrick would gleefully point out each time.)

#4: Basically an AIME problem. I feel like it's good, for now, for the USAMO to have a problem like this each year - something that has some overlap with the experience of those students who have focused solely on the AIME all year due to the fine filter of passing through to the USAMO. As I said earlier, more on this later. I do dig the quick proof of 'imagine there's a middle leg'. And you have to check out the limerick one of our community members came up with here.

#5: Nice problem. I like these 'no curricular math at all problems'. One of our Olympiad Problem Solving students came up with a solution that basically nails it with the extremal principle + the discrete continuity. Nice to see that there was some solution other than the convex hull argument, which MCrawford came up with faster than it took me to understand #6.

#6: Speaking of #6, this certainly fit the bill of a #6 in that it was fiendishly difficult to come up with a clean argument even after you understood basically what was going on. My main complaint with this as a problem is that it isn't easy to state (nor particularly interesting, though that's personal taste - MCrawford loved the problem).

All in all, I think it's a successful USAMO, though a tad heavy on the number theory and light on the algebra. I think there will be many fewer 0s-2s, and more students who felt like they had a fighting chance, all without making the test so easy that there will be a stack of perfect scores.