Here’s Mr. Suhaimi’s invitation to Justin, Yi Kye, Si Yu, Shazryl and I to participate APMO this year; only Shazryl declined due to exam matters. Yi Kye, Si Yu and I departed to Permata Pintar via taxi, and lodged at nearby hotel (thanks to IMO training committee for this treat!).
Three of us, upon arrival at the night before, received a pompous introduction by Mr Suhaimi. It’s timely for us to arrive just after certificate presentation session that we could bauble with juniors. Kin Aun, who stroke a consummate 10A+ in SPM, beseeching our opinion on costs and benefit of each scholarship, and condition in Sunway college; another junior who inquired my way of solving Problem 5 in last IMO after reading my post, and I had only vestigial memory on it. Out of eager and curiosity, we veterans attempted some test papers given in the camp and assignment problems, and the camp students rhapsodized how easy/difficult they were. How we wish we could return to those ages again!
Here comes the competition: 10 March 2015. Justin arrived at the exact moment when we settled down into our sitting positions, and were about to welter ourselves into the problems.
Click to access apmo2015_prb.pdf
Click to access apmo2015_sol.pdf
There’s only one geometry in the appetizer P1, which was a quick solve. Same for P2, with its brittle trickery stifled with diaphanous veil: once we beget the idea of playing with power of one prime, the solution was made crystallized.
Now comes problem 3, the only problem that I could be proud of for solving it. The final idea was like that of alternative solution 2 in the solution file inside, but I only managed to write it after 2 hours of treasure hunt. I viscerally tried the brute force method as in alternative solution 1, but the pattern isn’t “easy” to observe as professed. Indeed, just getting the “reversal” idea (which is clear since all terms are positive: just use a_(n-1)=(a_n -1)/2 for a_n >1 and a_(n-1)=2a_n /(1-a_n) for a_n<1.)
Then I tried small cases (even n) : some manipulation gave n=k for a_n=(2^k)-2, and n=2k for a_n=2^k. But for some cases, n=a_n (!) Is that a mutual perfidy towards member of foreign groups, or tryst among numbers of same family?
It too me until a_n=18 when things come to light (from above I once thought that n=2014 since 2014 is neither 2^k or 2^k -2): it was tedious to write it here, but final observation is exactly the same as that in alternative solutions: denominator+numerator=(a_n)+1 (first finding), and m_k=2m_(k+1) (second finding, in modulo a_(n)+1). Well, the final onus was to find order of 2 mod 2015, but all I needed to do was to write 2015=5*13*31, and prove that denominator must be 1. That’s all. Trust me: the things above was the summary of how this problem hauled me into bewilderment.
This left me 30 minutes left, and I didn’t find myself in good mood to solve the daunting P4 and P5. My ruffled mind lured me to eschew P4 and turned into P5–a problem that will get me nowhere. I did a simple observation of a_0, a_1, a_2, nothing else.
End of contest, few of us discussed how we did: 4 veteran olympians had a total of 11 problems in hand, a commendable performance given 8 months of respite. Another laudable one was Kin Aun with P2, P3, P4 solved (despite not finding order of 2 mod 2015, which was only a minor fissure). I was glad to see how he rose from last year IMO, and, proud of him as CLB successor (although, he had graduated technically)! I was especially stoned when he vaunted his success in finishing P4 in half an hour, and portrayed it as “easy”.
As the transports were scheduled at 3:30pm, Yi Kye, Si Yu, Justin and I had an extensive chat: how the mock exam was just 3 weeks away, how we were terrified by the exam that some of us even brought books there. Most importantly, how fearful was the idea of applying for university admissions: tedious work, yet low yield! Admission tests, essays, A-Levels,… everything counts.
Finally, three of us boarded the taxi and traveled into our own slumber land. “seems like three of you had bad sleep yesterday!” joked the taxi driver after sending us to our residence. Plausible, but false: we were jaded by the competition. However, being able to be here again after retirement was great experience, for we could return to the place where we had all our travails and laughter from IMO camps. How I wish I could be there again!