# [CIMC 2015 Part 2] Journey of the Blue-White Slippers

(Nontopical life update: Current 18.06 homework status: 34% (mildly screwed, probably won’t finish before I leave my cozy home for the U.S. and I usually struggle to get into the mood for homework while traveling, but I guess I’ll have to))

(I’ve been spending most of my uptime doing said homework and running errands, and my downtime catching up on Last Week Tonight with John Oliver while farming the Flight Rising Coliseum. And, okay, making the above status panel. Live version here courtesy of Dropbox’s Public folder. No regrets.)

### Day 3 (Excursions)

Morning routine snipped. We come to the middle school again to eat breakfast and gather; the contestants will be taking their tests here (accompanied by one bottle of “Buff” energy drink each) while the rest of us will be going on an excursion. Before this happens, though, two Taiwanese contestants ask me and Hsin-Po some math problems. There’s a geometry problem, which I fail to solve:

(paraphrased) In triangle △ABC, ∠A is 40° and ∠B is 60°. The angle bisector of ∠A meets BC at D; E is on AB such that ∠ADE is 30°. Find ∠DEC.

Hsin-Po figures out that, once you guess (ROT13) gur bgure boivbhf privna vf nyfb na natyr ovfrpgbe naq gurl vagrefrpg ng gur vapragre, lbh pna cebir vg ol pbafgehpgvat gur vapragre naq fubjvat sebz gur tvira natyr gung gurl vaqrrq pbvapvqr.1 Then, there’s a combinatorics problem in a book with a solution that they’re not sure about:

(paraphrased) 15 rays starting at the same point are drawn. What is the maximum number of pairs of rays that form obtuse angles?

This happens really close to the test starts and although I have this feeling it’s isomorphic to a notable combinatorial problem, I don’t manage to articulate the isomorphism until it’s too late and they have to go. Indeed, this is more or less equivalent to (ROT13) Ghena: gur tencu unf ab sbhe-pyvdhr naq n pbzcyrgr guerr-cnegvgr tencu vf pbafgehpgvoyr. After thinking though the solution on their book, though, I realize I’ve never seen this proof of said theorem before! (But later I realize it’s actually the just very first proof that Proofs from the BOOK offers. I probably skipped it because it involved induction as well as some algebraic manipulations that looked much less intuitive and natural than they really were, so it didn’t look as cool as the later proofs. Oooooops.)

I suspect I wouldn’t do too well if I had to participate in that contest right then. But anyway, excursion.

After a long bus ride, we arrive at our first destination, Jingyuetan (淨月潭 lit. Clear Moon Lake2), allegedly the sister lake to Taiwan’s own famous[citation needed] Sun Moon Lake. We tour the place on a wall-less car and look at the lake and lots of trees. During a stop, I take some pictures of sunflowers and bees, as well as a stand selling Taiwanese sausages.

The car blares weird music during the tour, such as a version of Für Elise with all the accents on different beats and a disjointed remix of the viral Chinese song 小蘋果 (Little Apple)3 with two other Chinese songs, connected with mumbling English rap segues. We also eat boxed lunches here while sitting on tiny, cramped foam mattresses on the dirt floor.

Our next stop is a museum, where there are lots of ancient historical artifacts I’m not very interested in. I find a collection of certificates involving or quoting Chairman Mao more intriguing:

But the phenomenon I take the most pictures of by far is the building’s weird combination of real and fake brick walls.

In addition to the noticeable difference in texture, the left side is solid if you knock on it, but the right side rings hollow. Sounds like it could make the perfect clue for resolving a locked room mystery with a little extra ingenuity somewhere. And look at these camouflaged panels!

We’re just missing a white panel with black brick texture lines painted on!

I know this is not at all the sort of thing that museum-goers are supposed to appreciate at museums — sorry not sorry.

Okay, so we return from the excursion and Cultural Night happens. The idea is that the contestants from the various teams will present performances to introduce their culture and promote intercultural exchanges during this night. This is accompanied by extravagant colored lights that are so bright, Hsin-Po and I both decide to put on sunglasses. There are way too many performances to discuss one by one, and most of them are rather unremarkable traditional folk dances, so here is a sharply abridged list of highlights:

• The very first performance is from the Netherlands. After a self-deprecatory spiel about their national dance moves and music, they got everybody to stand up, put their hands on the person to their left, and jump to Dutch music.
• Two countries, Australia and Canada, play games to introduce their country.

I confess, I didn’t know much about either and in particular for Canada I thought it would go like this part of the BuzzFeed video. Sorry, Canada.

• The best performances come from students and student groups of the middle/high school this event is being held in in the first place. This is perfectly understandable, though, since these students don’t also have to prepare for the math competition. By far the best of these performances is an a cappella performance of 天空之城 (Castle in the Sky), with Chinese lyrics that I didn’t know it had.4

I don’t know how to properly praise this. Even listening to my horrible-quality cameraphone recording gives me goosebumps. I would buy this song if it were on iTunes. Wow.

After that, Hsin-Po and I are semi-voluntarily snagged to grade papers. We find ourselves back at the meeting hall on the top floor of the leaders’ hotel, which is now much emptier compared to the leaders’ meeting yesterday. Mr. Li and a Bulgarian dude (among others?) are already working. Bulgarian dude hands us some grading schemes, which I am not very satisfied with — few alternate solutions are considered for some problems that sorely need them, and some solutions that produce and verify possibilities ex nihilio are partially credited as if they contained a proof rejecting the other possibilities. But I think everybody tacitly concedes there’s not enough time to regrade everything, so I just take them and roll with it.

These problems are actually not easy. Many of them are not easy because they involve lots of casework or scary computations, like multiplying two-digit numbers (oh no!)5 But some of them are hard because they’re geometry. (Remember, I’m bad at geometry.)

Elementary team contest problem 8:

E is a point inside triangle ABC such that AE = BE = BC and ∠ABE = ∠CBE = 20°. Find the measure, in degrees, of ∠CAE.

There is an elegant solution, although I’d be interested if you want to show me how to bash this with trig or complex numbers or barycentric coordinates too. Most of the papers for problems like this consist of calculating lots of trivial angles and getting nowhere, although I hear one person succeeds in trig bashing.

Some other interesting phenomena in the papers, this time combinatorial:

• Many people neglect the indistinguishability of counters on one of the combinatorics problems and don’t divide their final answer by 4, but have otherwise perfect solutions. This is sad. We only take away 3 out of 40 points for it, though.
• One paper uses an impressive four-set fifteen-case PIE. Fortunately many of the cases are clearly 0; unfortunately the paper forgot to undo an initial set-inversion and made more than a few mistakes in the nonzero cases. I think I ended up giving it 25/40.
• A lot of papers write $\binom{x}{1}$ instead of just x when counting something with x ways to do it. Is this a thing?

We grade papers through the night. At 3 AM I declare myself too fatigued to do more, so Dr. Sun gets me a hotel room here to wash and sleep. The layout is wonky — the toilet is in the already cramped shower stall here — but there’s nothing to be done about it. I shower and sleep until 8.

After the gang wakes me up, we eat breakfast and resume grading until the afternoon, while I keep an eye on the IOI Day 2 scoreboard. I’m kind of disappointed that it is really slow and occasionally suddenly requires reloading. Upon examination by Firebug I see `jquery.min.js` is grabbed from a Google API server and never cached; it’s a miracle it still works.

Minify and bundle your JavaScript, folks. It’s 2015. (Okay I’m being hypocritical.)

We actually make it with a few hours time to spare, but I am too tired to do anything except nap. Back in the hotel room, I nap from 3 to around 6 and awake with a foul taste in mouth. I feel highly debuffed.

Meanwhile, though, the others go shopping. Hsin-Po buys a weird pair of earphones, featuring Mickey Mouse ears jutting out at a clearly unergonomic angle from the part that you put into your ears, from a weird store called Miniso or Meiso, which looks kind of professional with a miniso.jp website but upon further examination appears to be a knock-off of various Japanese retailers that doesn’t exist in Japan. Bizarre-o.

After the above events and dinner, it’s time for actual coordination.

All the leaders are back in the classroom, yet again with one bottle of Buff each. Given my last experience in 2012 (which I didn’t blog about), which ended up in a loud argument over the point deduction for a trig bash that inverted a sine and cosine on both sides of an equation without justifying it with monotonicity, I am a bit tense, but it is uneventful. Most of the leaders describe useful partial results and request points that I am happy to give. There are only two remotely interesting incidents:

1. One leader wants to discuss a paper which does the first half of casework, stops there, and then suddenly produces the correct answer without explanation. I feel like giving 15 out of 20 (and consider it generous) but he wants 17 out of 20. We compromise at 16.
2. There’s a problem that goes like this:

Suppose a four-digit number is a perfect square, and if one subtracts the same value from each of its digits, the resulting four-digit number is still a perfect square. What is the sum of all such four-digit numbers? (Different values can be subtracted for different four-digit numbers.)

One paper just said that one could always subtract 0, and proceeded to (correctly) calculate the sum of all four-digit perfect squares as its answer. The leaders try very hard to get points for it. I don’t think they eventually got any, but I’m not sure.

I can sympathize with the contestant a little. Certainly, the phrasing could have been more rigorous. (But the phrasing I gave above was not the original English phrasing, which used “reduce” instead of “subtract”; I tried to translate it back from Chinese, which used 減去 as the key verb. I think that, in lieu of explicitly calling the value a positive integer, at least 減少 would have been better. In that sense, one could assign a little blame to the translators and the leaders for not catching this possible misreading.) And the IMC does not have any mechanism for submitting questions and asking for clarifications, which is also a bit unfortunate. Still, I think seeing “oh, we can subtract 0 so everything works” as the key insight to solving a problem ought to be a warning bell for reasonable contestants.

After our unexpectedly pleasant coordination and some number-crunching, Dr. Sun announces the medal cutoffs and prizes. The IMC gives gold : silver : bronze : merit : nothing in a 1:2:3:4:5 ratio, as well as some group and team awards. Finally, after midnight, we are shuttled back to our hotel to finally rest. And my shoes have dried so, after having had to wear them for two days, I can finally change out of my blue-white slippers!

That’s all for today.

1. ^ I think this method of solution — proving a property of point P by constructing a point P’ via a “nicer” route and then showing that P and P’ coincide, or sometimes doing this with segments or lines — is usually called 同一法 in Chinese olympiad terminology. Is there a corresponding English phrase? I have this feeling I knew of one at some point, but can’t remember it now.

(edit: yugrey says “phantom point”. That’s exactly it! Thanks.)

2. ^ Although nearly nobody calls it that. Sometimes it gets partially translated into Jingyue Lake. It doesn’t even have its own English Wikipedia article wow!

3. ^ Trivia: this was the song I played on the bus during IOI 2014 to “please my team and very strongly annoy one of them”, because it was released and went viral around the time of our training. Here is the YouTube link. It is weird. To put it mildly. If you don’t understand Chinese, don’t worry: it doesn’t make much more sense even if you do. It’s also actually an advertisement for a movie, somehow.

4. ^ Turns out that, as far as I can find online, it doesn’t. Instead, there are many unofficial Chinese translations that differ from each other in only a few phrases. The lyrics heard here appear to combine mutations from several of these versions, making it apparently unique; I cannot find an exact copy of them online. They are reproduced below:

Simplified:

传说在那天空上 在天际的远方 有座美丽的村庄 隐隐的飘浮云中央
谁愿伴我找真相 还要找到它方向 愿能有日向天空飞往 尽力找到它方向
探访天际的家乡 云层里是我家乡 寻觅着快乐温馨的土壤 心中的理想
在天空中有座城 有小野花飘香 在天空中有座城 鸟声似歌悠扬

傳說在那天空上 在天際的遠方 有座美麗的村莊 隱隱的飄浮雲中央
谁愿伴我找真相 還要找到它方向 願能有日向天空飛往 盡力找到它方向
探訪天際的家鄉 雲層裡是我家鄉 尋覓著快樂溫馨的土壤 心中的理想
在天空中有座城 有小野花飄香 在天空中有座城 鳥聲似歌悠揚

5. Mathematician: Sure. Does anybody know 12 plus 7?

Computer Scientist: You don’t?

Mathematician: What do I look like, a human calculator? Numbers are for children, half-wits, and bored cats.

— Ben Orlin, Math Experts Split the Check (I link to this way too often)