So this is a different numbering approach from chaotic_iak's. I may record lots of other details about stuff before the contest in my other blog sometime soon when I get around to it, possibly including a good deal of the rest of this post, but this is the actual math competition. Also it just occurred to me I never wrote about how I failed performed suboptimally at APMO 2012. I guess it's too late.

Score: I got the first two problems and the first part of the third (assuming no holes in my logic are found). Looks good.


This is a record of me taking the test, because each of us has to write a report on the IMO after we get home so I want to write this down before I forget anything.

The day before the test, as everybody was still slightly unsure whether we had adjusted to the time difference, most of our team took a nap. However it was just before this time that chaotic_iak came to my room. An extremely awkward conversation followed which ended in me awkwardly closing the door and headbashing behind it. As a result I barely slept because I was mentally conducting what can be frighteningly accurately described as a post-mortem analysis of the conversation. 1/3 of the remaining free time that evening was spent practicing conversation skills with the wall. Mehhh.

And then after procrastinating with a variety of other things, I went to sleep at 11 PM and woke up at 3:30. Something is seriously wrong with my biological clock. Of course I made myself stay in bed daydreaming until about 6:30. Breakfast, gathered writing utensils, set off to the contest hall.
The tables here are smaller than they were last year. Also the proctors barely checked what people were bringing in and out in their bags. Last year everybody's envelopes were checked one by one and there was even a metal detector. I hope nobody tries to do anything. Anyway, I tried to calm myself down with breathing exercises and stuff until I heard the starting signal, and opened up the problem envelope...

Problem 1: geometry. I was ambivalent about this one, because although I would probably have the highest chance of failing this sort of #1, I didn't have too much difficulty with easy geometry during practice, and it opened up space for combinatorics in other places.
Problem 2: algebra, in fact an inequality... uh-oh. I hate inequalities (as does the rest of our team, really). This one didn't even have an equality sign; it seemed it would be troublingly nonroutine. What the heck?
Problem 3: combinatorics... awesomesauce, now I have a shot at 777ing today!

Right off the bat, extremely conflicting emotions.

All the panicky anticipation aside, Problem 1 was smooth sailing for me, taking thirty-five minutes. Having read the other solutions on AoPS, though, I now realize mine is incredibly long-winding. I think I invoked Menelaus completely unnecessarily; I'm too lazy to check now. Geometry is still not my forte.

I think I decided to try #3 first because scary inequalities are scary. For the first half of the problem, I struggled for a bit trying and got a messy strategy that didn't generalize at all, and after a bit more work, checked out #2.

I expanded #2 for and couldn't do anything; my best attempt was just throwing AM-GMs together and it led nowhere. Rather troubling. I took a couple logs looking for Jensen and some other things, but the lack of an equality case kept bothering me. Finally I gave up looking for elementary solutions and tried to bash it with Lagrange multipliers, probably for the first time in over a year. Luckily (or not) I had reviewed it on Wikipedia just a couple days before. It was actually rather straightforward to reduce everything to one inequality in one (auxiliary) variable with one constraint, but it was still impossible for me to see why the inequality would hold.

Back to #3... I came up with something, started writing my solution, realized it didn't work but that was okay because there was a lot of preparatory work that was still usable, fiddled around and came up with my new strategy. At first I thought it was too simple to work and that it would contradict the second half of the problem, but after looking for quite a while for a mistake, I convinced myself that it worked. I finished the write-up for #3.1 (posted here in the IMO forum). About an hour left, and I was now faced with a big decision: do I shoot for the second half? Do I go back and try #2? Or do I try to multitask and hope that at least one works?

I went back to #2. I realized that it was only a freaking #2, last year was severely anomalous and even then the solution was simple, there had to be a simple solution using something easy... say, AM-GM. Suddenly I realized how I could cut up the constant for AM-GM in order to get a clean variable term of after applying all the exponents... and what do you know, everything else telescoped perfectly into the RHS. I facepalmed a couple times. I wrote down the solution, one spacious page compared to the bulky ten pages of scrappy bashing behind it. I facepalmed a couple more times.

There were only maybe twenty minutes left, so after paranoidly checking everything I had written and jotting down all the header information as quickly as possible, I scribbled some intuitive shots in the dark for #3.2.

Day 1 was over.