Tournament of Towns 2011

Last Sunday and this Sunday. Yup.

I’ve been skimping on olympiad practice lately, what with all the hospital visits, homework, increased AoPS forum-gaming, and so on, so I was slightly apprehensive. All the apprehension disappeared after writing the O-level paper, though. I skipped straight to #5, a pure combinatorics problem posed with simple geometry, and was slightly annoyed and surprised to find a very routine problem, not as interesting as I expected from TT, and not the difficulty I had expected.

At first I thought maybe the problems were just not ordered very well (windmill *cringe*) but then after I had finished it I discovered that all the other problems, including the pure-geometry one, were not even routine, but trivial. I guess three months of training doesn’t just go away.
1. Inequality loops and a parity check
2. Just take the diagonal!
3. Bash, with e.g. Law of Cosines (opposite sides’ squares have the same sum)
4. Push a few inequalities with divisibility

My proof-writing skills have become rather rusty, though; or maybe I should just go back to using English to answer as well. This paper had the unfortunate effect of making me a little complacent during the A-level paper, though.

Once again, the hardest problem was pure combinatorics in a geometric background. This was definitely not routine, and I think I used too much time on it without completing it. I fakesolved this problem twice and used lots of time writing incomplete solutions, although the second try made some nontrivial progress, I think. It was easy to keep discovering contrived conditions that a theoretical counterexample would have, but a contradiction was still out of reach.

The first I completed was problem 5, pure and fun algorithmic combinatorics, which I also kind of fakesolved halfway before figuring out the surprising versatility of the operations (although surprising versatility really should be expected in fun algorithmic combinatorics). Then I used lots of time trying to get at 7 again, randomly taking breaks to manage to get 6, 4, and (just to be safe) 1.

6 was NT, and not easy; luckily, power-of-two moduli and their lack of primitive roots were still fresh in my mind from that problem james4l solved from chaotic_iak, and it fell. 4 was pure geometry, just bouncing around cyclic quads produced from the orthocentric configuration, although I was too lazy to write the proof with all my usual rigor. 1 is probably safely combinatorial geometry and was just length-chasing.

And… that concludes olympiad math competitions for four more months, until the APMO rolls around.

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