IMO 2014 (V)-The climactic section (Second Half)

Things were essentially the same as before. This meant, no watching of FIFA semi-finals the night before to ensure good rest, and early rise and shine in the morning to catch the Jamie Shuttle. The only difference was, we were only allowed to enter the hall 15 minutes before the paper, in which my meditation plan was cut short.

On the other hand, on speculations on topics, P4 would almost certain to be a geometry problem, considering the power of country leaders with extreme favor on geometry. It was expected that P5 would be number theory if the protocol of Problem 1,2,4,5 under different topics was applied again after its first introduction last year. Problem 6 was likely to be algebra to complement P1. I hoped for GAN instead of GNA, though, since I was stronger in number theory compared to algebra.


An astonishment arose in me when it was GNC! Problem 6 looked unusual compared to others when it offered partial credits for weaker results than what you need to prove. With my deficiency in combinatorial problem solving, I virtually gave up the aim on solving all 3 for the day, leaving me lower chance for a gold medal.

P4. With hands trembling for cold, I picked up my compasses and constructed the diagram. The intersection, R, and triangle ABC seemed to have extraordinary relationship: ABRC looked like a harmonic quadrilateral. While the problem wasn’t immediate to be solved, it was natural that ABC, PAB, QAC are similar. This, suggested the idea of reflecting B, C in A to form X, Y. The light of hope was there when MAB and YCB are similar, so do NAC and XBC. And see the parallelogram? Angles XCB and YBC are supplementary, and so do angles ABR and ACR. The doubt was cleared at 9:06, and with full solution created at 9:19–a great improvement from last year.

Before jumping to any other problems, it should be noted that time for queries was still open until 9:30, so I read the problems again and found that P6 itself was questionable: can a region containing several finite regions considered as a finite regions too? Simple logical thinking negated it, since taking any three blue lines would then refute the problem. Nevertheless, there was no harm in submitting queries, and the reply was consistent to my guess.

P5. Does 100 have any relation to 0.5 in the problem? No, thus let’s replace 100 and 99.5 with n and n-0.5, and proceed by induction. Immediate assumptions made: no coins with total value exactly 1, at most k-1 coins with value k, and consequently we can assume that there were at least n+1 different numerical values of coins. The existence of coins with value at most 1/(2n+1) would be advantageous, since by induction hypothesis take any pile with at most 1-1/2(n-1) (the average) and this would settle the problem instantly. But how could it be true?

With this handicap, I experimented with the smallest non-trivial n again. For n=2 we may need to consider if any of the coins was less than 1/5, or not. I then realised a trick: how about adding 1/2+1/4+1/4? Hmm…what could 2/4=1/2 and 1/2+1/2=1 suggest? The merging of coins, of course! And here’s the miracle: coagulating the coins together to form a coin of 1/m for some m would give another simpler configuration, yet giving the true conclusion. I knew what to do now, and in the end we had at most one coin of 1/2n (as 2/2n=1/n). That was indeed a shortcut in leading to success: as (2n-2)/(2n-1)+1/2n is less than one, these could be combined into a valid pile, with 0+1/2=1/2: extra space for those smaller coins with total value of no more than 1/2 (each of less than 1/2n) to fit into it.

But what about if the total value of small coins was more than half? This beneficial condition enables the proof by induction, by creating a pile of value slightly more than one, and exclude one small coin (value 1/(2n+1) or less) to make it as valid pile. Then by induction hypothesis, there is n-1 piles with one of them having value at most (2n-1)/2n so we can easily add the “small coin” in. Simple right? Unfortunately I consumed a lot of time in it, leaving me 2 hours for the other problem.

(Ps: Anyone noticed the artificial combinatorial flavor in it? In fact, that scent was stronger than that of number theory. Some uproar happened in AoPS when it turned out to be derivative, and juries were consequently denounced in the forum.)

P6. The moment I read the problem, I felt abject: how on earth were we going to do that? Visualizing the finite regions was itself some challenge, where the first thing sipped into me was a configuration of plant cells. Well, I could have made my life easy by waiting for time to past, or being productively, checking my solutions for P4 and P5 (and I did check!). But, no. I attempted to make use of my “creative” imagination to bestow my readers with facts directly apparent in front of you, like “each finite region must be convex” (with outline of proof) and “if there is a convex n-gon then the claim is obvious” (I doubted whether I proved it). This was analogous to say “enjoy my writing of nonsense”.

Whether c square root n worth a mark was immaterial, since I didn’t touch anything about it! (Hahahahahahahaha)

It seemed like six of us just went through a stormy voyage and returned with torn clothes. P4 was solved by 5 of us, with another one reducing the problem into the condition AB/BC=AR/CR. P6, as expected, slaughtered us into fine pieces, where Zi Song unabashedly wrote “the problem was obvious for c=0”. The real tragedy was P5, ironically, since no one else held it well in his hands. I assured them that P5 wasn’t easy, where the short problem statement belied its difficulty. Two of them made some partial progress, expecting some scattered points as return. We nevertheless forgot about the predicament of the day, and joked about the constant value of c in P6. “Does proving c>1 give more than 7?” This amused us.

“The IMO had just started-How Si Yu”
Indeed, above was the “happy-go-lucky” quote from Si Yu after the Day 2 contest in Colombia, last year. There was no use to cry over split milk, so why should we spend our energy thinking about it? Most importantly, we met our father of team: Mr. Suhaimi, the leader of Malaysian IMO team. Although we had limited time for conversation (unlike last year’s verbose conversation discussing on IMO problems), that was sufficient to give us a feel of reunion. That was just before our plan of entering game room to be engaged on bicycle (tradition retrieved!) with some Moldovian team members.

Excursion #0-The Dance Floor
The official excursion program began on the next day, but owing to the meticulous plan of IMO 2014 organisers we were given “excursion” to the musical world: teaching of African Dance using a song “Shosholoza” (Move Fast). The song reflects the immense passion of South African working under the mine, despite facing tonnes of affliction.

To start with, we were taught rhythms on drum tapping and tambourine playing, and consequently we were asked to perform in front of public–the so-called “sight playing”.

Video of the song:

Drummers of the day
Drummers of the day

Lacking the ability of dancing, this was what we did when we were given choices between singing and dancing:


Fever on feet
Since we missed the last match, we wished to fulfill our craze (or, curiosity) by watching the match of Argentina to Holland as semi-finals. However, there was practically no excitement in watching it some distance away with crowds in front of you, forcing us to choose the alternative option: Bicycles (again).

This concluded Day 2/ Pre-excursion day.

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