# [IMO 2012 Part 6] Mostly Not About Excursions

Yes. I know it’s been more than a month. Blogging motivation decreases, but the responsibility of that stay tuned doesn’t go away.

It’s okay. It’s all worth it because the stuff in the games room is absolutely ridiculous. Warning: huge post.

Our old friend, the monster of a functional equation, in edible form. In the games room. Did I mention abso-zarking-lutely ridiculous?

# [IMO 2012 Part 5] Unlucky Fours

[edit: okay guys I’m surprised at many people come here with search queries looking for solutions. If you want IMO solutions, the corresponding AoPS forum invariably has many of them. This is probably late-ish, but just in case.]

Day 2 of the contest.

Did you know that in Chinese [or Mandarin, whatever] “four” is unlucky because it’s a homophone for “death”, and hospitals tend to skip it in floors or ward numbers?

Did you know that there was going to be an anecdote involving the seventh Artemis Fowl book but I couldn’t make it work so instead you have a weird and utterly disconnected metareference to something deleted?

I don’t know, it sounded cool at the time.

Problem 4. Find all functions f: Z -> Z such that, for all integers a,b,c that satisfy a+b+c = 0, the following equality holds:
$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$.
(Here Z denotes the set of integers.)

An innocent-looking functional equation, but once you start trying it you discover that there’s quite some depth to it. Random guessing can yield that $f(x) = x^2$ is a solution, so I proved inductively that after dividing out a constant f(1) then the remaining part of f is a perfect square. Letting $f(x) = f(1)g(x)^2$ with g(x) a nonnegative integer and factorizing the original equation, I got an auxiliary functional equation equivalent to the original equation.

Casework on small values of g, and the surprises started coming hard and fast. First: wow there’s an extra odd-even solution! Then: woah there’s another mod 4 solution! What is this madness?

# [IMO 2012 Part 4] The Inequality of Broken Dreams

I could spend all day coming up with post titles like this one. Really, I could. Anyway, after receiving the letter gently reminding me to turn in the official IMO report in Chinese, I finished that first and turned it in. This, plus the fact that I don’t have any way to do any of my summer homework yet, should make continued blogging much easier.

These months of preparation and anticipation, and in just over two days it will be behind us forever.

Thus were my thoughts at 3:30 in the morning, which were a much worse method of preparation than that which I used for the APMO, which involved trying for the first time to put my compass in its box and staring at a cryptic crossword. Twisted thoughts as I lay innocently in bed, trying to preserve my spirit and mathematical function. I got up at 6:30 and ate a breakfast nervously with the team.

We dispersed into the contest hall. Flat squarish white tables in orderly rows greeted us. They were considerably smaller than the arm-span tables we had last year in the Netherlands’ stadium, but the space was still ample compared to the ones we took our practice tests on at home. I floundered a bit looking for whoever was supposed to check our stuff for forbidden items, but none of the proctors paid any attention to that, so I flipped through my own jacket pockets paranoidly.

But preparation was too brief, and there were at least fifteen minutes left with nothing to do except try not to panic. I managed to fill these fifteen minutes by doing elaborate breathing exercises, raising and flapping my arms, counting down seconds, focusing on a color word, meditating badly, clearing out everything nonmathematical from my mind with a symbolic gesture. This is not a very coherent description, but I wasn’t feeling very coherent.

I opened an eye halfway and watched as the clock ticked down the last few seconds. The starting signal sounded. I opened up the problem envelope…

# [IMO 2012 Part 3] Luggage Carriers and Language Barriers

Who knew coming up with strangely cryptic and illogical but nice-sounding post titles was so much fun!?

Okay, this break in storytelling is mainly brought to you by generic summer laziness, as well as possibly a tiny bit of chemotherapy adverse effect. There is only a little perfectionism involved, and that’s because this post contains a long awkward situation (you might have guessed already). I guess this is what it feels like to have oodles of stuff to blog about, but not enough motivation. Heck, these few days with running the TAIMC have long filled my list of rant topics with juicy stories until it’s near-bursting. But I need to end this self-pity party before I get carried away, so… back to the action.

Stranded in a foreign country at least 20 degrees Celsius below our comfort zone, having worn the same clothes for 36 hours of airplane travel, and still with less than 48 hours until the contest and zero out of six compasses, we were running out of options.

A quick inventory of clothes showed that, after including the vests and caps in the backpacks we got, we probably had just enough clothes to survive the cruel frigid environment. So, reluctantly, we left the games room and hit the street to hunt down some underwear and socks to change into. Not to mention toothbrushes and some T-shirts and jackets for good measure, because the inside of the hotel was not the right temperature for full cold-resistance gear.

Some haphazard wandering up and down the streets later, we found a store that suited our needs and picked up some clothes. The underwear came in two sizes: too loose and too tight. We picked the latter. Oh well, it would only be for a day or so… right?

# [IMO 2012 Part 2] The Hotel of Deception

We finally arrived at the hotel at 3:30, meeting another local from Taiwan, Mr. Chen, who helped us carry some of our stuff off the bus. Po-Chiang, our guide, was waiting inside. We took more pictures and finally lugged the meager stuff we had off to our hotel rooms.

At least, we tried. I started to realize that there was much more to this hotel than it seemed.

Firstly, of course, was the confusing placement of rooms with numbers starting with 4 and 5 on the fourth floor (which would be the fifth floor by our numbering system, where the lobby is floor 1; but here the lobby was assigned 0. Off-by-one errors just waiting to happen here.) Secondly were the completely indecipherable signs. I don’t remember the details, but the first signs we saw read something like “560 ~ 540: left; 520 ~ 540: right”. Occasionally there would be weird slashes or half-slashes between the numbers instead (later I finally realized they were slanted, Comic-Sans-style capital Ys, or “and” in Spanish). Are these closed, open, or half-open intervals? And why the heck are their upper and lower bounds in a different order!?

We wandered through the corridors, peeking down each one, trying to figure out whether the numbers were increasing or decreasing and whether a parity argument (for those of you not fluent in math lingo, that means odds and evens) allowed for the existence of our room. Who knew the simple act of finding one’s living quarters could be so mathematically tasking? In the end, our rooms were in the last corridor, just about diametrically opposite to the elevators on the half of our floor. Oh well.

The room was pretty nice overall. The furniture and basic facilities were quite complete, with a sparkly bathroom and a couple tables and chairs of various shapes. The closet was big and had a safe, which was rather important because just about everybody we had met had warned us over and over again about all the incredibly skilled thieves, muggers, and pickpockets in Argentina. It was probably much safer (no pun intended) in the hotel, but with all of these warnings (later we would even find a notice from the hotel warning us to lock our doors) I was never entirely certain. There were lots of lights controlled by a set of confusing switches on either side of our beds. There was at least one white immovable divider cunningly disguised as a switch, one switch that didn’t ever seem to do anything, and one that turned everything off. The last one made a little sense after a while because it had pictures of stars and a moon on it, but the whole setup was still pretty non-user-friendly in my opinion.