[IMO 2012 Post 4] Six Problems

Hm I start to forget stuffs in IMO. Better get these out before I forget it entirely.

In short, I did extremely poorly in it, at least for my target of getting 21 points or silver whichever is higher. Stuffs aside, I’m having moar fun in the recreation hall, especially after Day 2’s contest where the stuffs of recreation hall becomes available.

Day 1. Going to my desk, I found out that it’s smaller than the one we used for mock IMOs in our training. Combinatorics thoughts (read: how to get more working space) crossed my mind, and I came up with a fairly effective way to enlarge my working space by stuffs including putting my…envelope or whatever standing in between two bars that border my desk, stacking the snacks on top of each other, and more things. It turned out that I eventually only use the working space of three papers of whatever size that is, while the desk has like twice of the amount of that. Oh well, at least I can do packing quite well.

That aside, let’s go onto the problems.

Err I have covered it here, I believe. Okay, let’s just go over that again with greater detail.

First, of course I read everything. Geometry, Algebra, Combinatorics. Exactly as I predicted, but the results were practically the reverse of what I expected (I hoped for 777 given that prediction because I should be able to tackle 1, I’m pretty good at Algebra to should be able to attack it even at 2, and I’m great at Combinatorics that I should be able to slam one on 3–wait why am I using Pokemon attacks).

The combo was a huge text wall that took me one minute or so to read and comprehend. Seeing a possible ambiguity in Problem 3.2 (whether we need to ensure such n exists for each time there is an integer between 1.99^k and 2^k or we just need to ensure such n exists for every k greater than something), I rapidly wrote a question asking that. Afterward, move to the geo.

It was simple angle chasing (after I redid the problem at home). However, due to pressure or whatever, I failed to see the concyclicity of ALJB (\angle AJB = \angle ALB) during the contest, which basically cost me three points (I eventually got 4 by doing the angle chasing up to that point). Trying to trigon bash it, I spent way more time than necessary. I left it for the inequality.

Seeing no way to break the exponents, I went straight to derivation. Fixing all but two of the a_i’s, I come up with a condition that must be satisfied by these two variable a_i such that the result is minimum. Hence, the minimum is achieved when all a_i satisfy this condition, which I then wanted to prove to still be larger than n^n. However, soon enough I couldn’t find how to plug in those. Then how can I expect to find its minimum…

The combo was basically extremely weird; I didn’t get any idea at all. I suppose I must practice doing game theory with holding 2^n + 1 states. Also, I did an induction to prove that B only needs to have a winning strategy for 2^n + 1, but the induction has a flaw, costing the possible point I might get there out.

Afterward, I just flipped focusing between the three problems, failing to get anything else.

Meh, so I expected 231 there. I eventually got 400, worse than I thought (expecting 6 points, got 4 points). So much for getting a perfect prediction.

Let’s go to Day 2 first before returning to Day 1’s night.

I expected Number Theory, Combinatorics, Algebra because again, I should be able to tackle 4, Combinatorics in 5 is basically free 7 points (I managed to solve last year’s C5 in shortlist, and C6 should be Problem 3/6), and Algebra at 6 should give at least 3 points. Expecting 773 if I get that. Apparently it’s Algebra, Geometry, Number Theory; completely incorrect prediction. But whatever.

The algebra was a very easy functional equation that should be free 7 points. But I did an absolutely stupid mistake when rewriting the solution; if I didn’t submit my scratch, I think I will only get 1 point or something. Luckily I have a perfect solution (only really messy, hence why I rewrote it) in the scratch, giving 6 points. But then no HM is possible, since I didn’t solve 5 and 6 at all. I tried coordinate bash for 5, which gave many things but finding M became extremely scary with the coordinates of K and L being two lines each. I only got n = 1,2,5,6 for 6, so I claimed these are the only possible solutions (which was apparently incorrect since n = 1,2 mod 4 was the answer).

I expected 410 after working on them. I got 600, slightly better. But that netted only 10 points; I’m pretty sure I won’t get bronze, and HM is obviously out.

Let’s back to Day 1’s night.

Err… Actually nothing notable. Insert usual dinner + recreation hall. I think I chatted with betaveros again this night, with me still being difficult comprehending his speech. Or is it the next night? I forgot.

Day 2’s night, I see an extreme improvement in recreation hall, with the most notable I remember being Laser Quest, tournaments, and pool tables. However, the next day (Friday) is the day where I spent a lot of time trying new things. That’s next post.


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