Again another contest done: this one lasted for 11 days. With Zi Song spearheading, Justin, Yi Kye and I joined this post-IMO math festival.
Click to access OMOFall14Probs.pdf
After much derail of thoughts, getting over onerous algebraic expressions, scouring each and every little case, we managed to score 27/30, leaving ourselves thrashed by problems 27, 28, 30. We then finished #9, a somewhat respectable position at this international level.
I happened to know one of the members in the premier team, Cody Johnson. He seemed to hanker for a position in IMO team, which I believe he would make it eventually. Meanwhile, another quasi-Malaysian team took part, and scored #36.
Comments on selected problems:
Problem 15: It looked like Fibonacci series, prima facie, hence being too profound to find the pattern, until…we found a simple induction method, and bounding of numbers.
Problem 18: Hermite identity didn’t penetrate my mind straight, and I did the rearranging part of it (as in the solution), reaching a stalemate. It’s induction’s power which prevails and lead into the almighty Hermite identity, again.
Problem 21: Wait, what? Let x_2 to be x and instantly you get x_5=x_7. This lead me into solving a (if not mistaken) quintic equation, which, by definition, might not be solvable. What’s more was, Zi Song debunked my proposed equation, and used brute force approach. When you look at the official solution, you go into apoplexy, stunned at the proposer’s insight.
Problem 23: Oh yeah, Lucas’ theorem that got into oblivion. Although I obtained the same result (2^odd=1), I tried in really hard way, expressing every NcR into N!/R!(N-R)! The terms as in Lucas’ theorem appeared, but I was befuddled by the (-1)^k (as of Wilson’s theorem) after removing all powers of p. In the end, it’s good. They cancel each other as of parity.
Problem 24: Shhh……we took the fact that regular simplex being symmetric for granted. (Just because this is tacit from the definition?)
Problem 25: Open confession: the only problem that we guessed the correct answer. How? I tried out polygons with 3,4,5,6 sides and get (n-1)^2+(n-4), and reduced the sum into (n-1)^2-1+o-e, where o=odd sided convex polygon with exactly a point in it (similarly for e). As I went on, all work was menial and I could not coin any idea from it. Thanks to Justin whose guess corroborated with mine, and claimed a configuration from it.
Problems 26 and 29: trolling geometry, with the former demanding the flare of proving that incentre of ABC is perpendicular from X to MN, and latter artfully hiding the fact that I, E, M being collinear. Sorry, but with presence of Justin the disguising artifice of problem was obsolete.
Final three that overthrew the IMO numerators.
Problem 27.: I went into euphoria went I claimed a solution, until I read the problem again and realised that f can take value zero. For the next few days, my mind belaboured around finding whether sqrt(2)+2 is a quadratic residue of p. I’m not the only one: Zi Song was in the same boat with me. I made myself simplistic, and assumed that it is not a quadratic residue (hence the team guessed -4=65533). When we checked AoPS the day after the deadline, we found ourselves drowned by the malevolent boat.
Problem 28: differentiation, writing b in terms of a, brute force,…all permeated my mind. Justin and I both obtained a 8th degree polynomial (again not directly solvable). And now you hope: if only they allowed complex solutions. Zi Song simply alleged a 3, proximal to the official answer, in turn.
Problem 30: finally, this one. Telescoping 2nCn(4^(p-n)) gives 4(-1)^(n) ((p-1)/2)C(n), but H_n is inflicting when it confronted me with reciprocal series which seemed extraneous to congruence. I mulled on it for some time, enjoyed myself as a buff for this problem, and hoped for some positive return. And the screen response “sorry, this is not your lucky problem”.
Overall, we had fun on playing with this paper. As IMO came to a complete stop for me, I occasionally found myself dwelling on all beautiful moments in the 4-year-journey: a period that I could not return physically. Now I got to do math again (this could explain my notion on writing this article). Finally, as cliché as it could be, thanks to my teammates: Zi Song, Yi Kye (who apportioned time from AS studying regimen to this OMO) and Justin. (But does Yi Kye need to study at all? 😛 )