Evolution in maths

#randomthoughts #RIPgeometry

Last July (a month and a half before I departed to Waterloo), I had the privilege to attend an Expii seminar hosted by its founder, Prof. Po-Shen Loh (also the leader of the IMO Team USA which won two consecutive IMOs, and the head coach of Team CMU to the Putnam 2016 which finished first). The talk itself was refreshing, with descriptions of a unique approach of teaching maths  (with a demonstration of solving a puzzle involving 8 x 8 table efficiently).

The gist of the day, nevertheless, lies in the after session (courtesy to Mr. Yeoh and Mr. Eng, the fathers of two students into the IMO camp).

Since it was right after the conclusion of the 2016 IMO, I started off, “why is there only one geometry problem in this IMO?” and Prof. Loh replied, “why must there be two?” I was stunned: this is something that I always hoped for but I myself couldn’t tell why. He continued by saying that the team was quite ‘green’ (i.e. new to math olympiad) and the members won “only because the problems are rather unconventional”. I hardy recalled the exact content of the ensuing conversation, but I could attest that he was a proponent of getting non-traditional problems onto the paper: that was how the topic of evolution and change started. At one point we delved into the aspect of transportation, which I said how taxi drivers are disgruntled by the proliferation of ride-sharing services like Uber and Grab. He continued by something stronger: “who knows if these ride-sharing drivers will be replaced by autonomous cars in the future?” An interesting thought, indeed.

The after-session

As I thought further, one reason that I am an avid fan of geometry-heavy math Olympiad paper was my longing for the perpetuation of status quo. When I first stepped into the IMO training camp, Mr. Suhaimi talked about the sacrosanct position of geometry in the heart of mathematicians as people started understanding mathematics through geometric diagrams and partition of lands, “you should therefore love geometry to ace the IMO”, he continued, which indirectly inculcated the importance of geometry in our minds (disclaimer: that’s during the old good days when we had two geometry problems in each IMO contest).  My passion for geometry also comes from my relative strength on geometry compared to other topics, especially after my unexpected (magical, rather) discovery of trigonometric hacks in solving geometry problem. It’s just heart-wrenching to see my strongest and favorite subject being ruthlessly de-emphasized in the IMO.

Further explaining people’s fondness towards mathematics was the relative simplicity to train and excel in geometry compared to other topics. Indeed, to quote from a former IMO teammate (Ying Hong), “you will find yourselves breaking a math problem into a few smaller sub-problems, and finding them is itself a challenge. In geometry, finding the sub-problems are way easier”.

The discussion of conventionality extends beyond geometry itself. In inequality, for example, a classical trick is to use Muirhead’s inequality which required little ingenuity, albeit being more tedious due to terms expansion. That’s another reason why inequality today aren’t that solvable using Muirhead’s: as quoted from this Inequalities handout by David Arthur, juries are aware of the vulnerability of certain problems under this inequalities and will therefore avoid them. On the other hand, some other classical tricks like Cauchy-Shwarz are still being favoured: it requires more ingenuity to notice that an inequality problem can be vanquished by this trick, with some tweaking of terms necessary.

A serious question to ponder would be, given the types of problems favoured by the juries these days, how to excel in math Olympiad from time to time? (Well this question might not be relevant to me, but still worth a thought). In hindsight, an inherent weakness of myself in my math Olympiad journey was that my performance depended significantly on the types of problems given–especially in real contests when time constraint matters. My performance could fluctuate from solving IMOSL 2013 N6 in two hours to merely nailing problems 1 and 4 on the Spring 2014 Tournament Of Towns A-Levels (I have to concede that TOT isn’t really a contest I could easily thrive in compared to APMO and the then-IMO). How could legendary individuals like Alex Song, Lim Jeck and Lisa Sauermann kept vanquishing the IMO problems regardless of how bizarre they were? The key is, probably, to embrace the new types of mathematical tricks from time to time. This is easier said than done, but resembles a requisite for one to call themselves a true mathematician–to exhibit interest in every field in mathematics that is getting increasingly diverse every day.

Let me end this post with a question a friend asked me: “do you think math defines the world or world defines math?” Here’s my answer, edited for clarity:
“The answer is both. Math defines the world because some objects in the nature has heavy mathematical components in it (like the number of petals being Fibonacci number, and beehives being hexagonal). The world is determined by people, and sometimes it’s the people themselves who determine which types of math we do. E.g. look at the 2017 IMO, in which the dearth of geometry problems on Day 1 and the novel nature of Problem 6 suggests a high level of unconventionality.”

PS: let’s not forget the geometry diagrams that are aesthetically pleasing to eyes while cracking the problems. You shall be remembered.

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