To begin at the end...

Hey!

I haven't posted for a long time, and that's largely because I don't know where to begin. My tryst with Olympiads began in 2002 and still continues, though my role and position with respect to them has changed continuously. And catching up with four year's worth of stuff is daunting, and there have been other things to do, so I've been putting it off for quite long.

But now I have a place to begin with: today. It has been two years since I went for my last IMO, but my interaction with the subject hasn't ended there. Last year, I went for 2-3 days at the end of the IMO training camp at HBCSE and gave a medalist's session. This year, since I was anyway in TIFR Mumbai, I visited the Pre Departure Camp (meant solely for the team), again in HBCSE. So I'll begin by describing that experience, and may be that'll help me get into rewind mode myself.

So for the mundanities. I left TIFR at 6:30, reached HBCSE at 8:00 a.m. (BEST bus 21 LTD). The team of course hadn't turned up, so I entertained myself for an hour before I saw two of the "team members", Apurv and Riddhipratim, turn up for breakfast.

Chatting with the team members began. It was the usual list of questions: questions about how CMI, my current institute of study, is for B.Sc., and how it compares to the B.Math. programme at ISI Bangalore. CMI and ISI Bangalore are widely recognized as two decent places to do an undergraduate degree in mathematics, within India. The question of which one is better is an important one for anybody keen on pursuing mathematics. That's just like the question of which IIT should I join haunts IIT qualifiers and which US university to join haunts people who make it to a large number of similarly graded universities.

I've often thought that information in this regard is rather sparse... in fact, it is only the privileged few who have even heard of CMI and ISI. And even these privileged few don't really know what goes on in either place. But what's the way out. Informative as the websites may be, they can only go so far. May be I'll put in some blogs on CMI at some point in time which will help people be better informed on whether or not to join CMI. Though of course, you'll have to keep in mind that I have a personal stake considering that I am from CMI.

But that's an aside... I asked the team how they were "preparing" for the Olympiad. An international Olympiad, as you might have guessed, is more than just a mathematics contest. It is an event for people from different countries to interact with each other, it is an occasion for visiting and exposing oneself to the cultures of other countries... it is a very special kind of occasion that can be irritating at times (specially in terms of the food and when the excursions stretch a little too long)... but on the whole, very much worthwhile. The team confessed that they didn't even remember the name of the language spoken in Slovenia, the country where IMO 2006 is going to be held. I told them to figure these things out... we had faced our share of problems in Tokyo (IMO 2003) and Greece (IMO 2004).

This time's team comprises six boys, breaking a four year long trend of one girl in the team. Never mind, there are ups and downs in gender proportions.

When breakfast came to an end at 10:00, we decided to move over to the lecture hall. The teachers said I could spend the day teaching and giving tips to the students the way I wanted. Before launching off, I asked the students if they had any general questions about me. One guy, Varun Jog, asked me how the interaction between teams had been. I said that regettably, it had not been much, but the new team would hopefully interact more... the IMO is an opportunity to understand other countries.

Ever the practical and experienced Olympiad goer, Abhishek Dang pointed out that it wouldn't be quite possible for us to barge into the rooms of other teams at night. "Why not?" I asked. This is an important opportunity that we don't get very often. I also told them to feel free to go to the dance floor, something I didn't do in my own time :)

Then began my first serious lecture. This was in number theory. To tell the truth, I hadn't really prepared to give any lectures, but I've been trying on and off to prepare Olympiad materials, and I just talked on the themes that I had been sort of working on. The first one was on "number theory". I did a little about modular arithmetic, introduced terminology such as "group", "ring" and "field" and plunged into a number of problem solving tricks. I hope to post an article on my Olympiad page soon, but I'll still jot down the important points I made there

(i) The congruence classes
(ii) The fact that if p divides a^k - 1 then the order of a modulo p divides the gcd of k and p-1.
I gave many examples of this in different forms: the least prime divisor method, the cyclotomic polynomials, Mersenne numbers, and Fermat numbers. (All these are discussed in detail in my upcoming article).
(iii) The concept of field extensions and the general version of Fermat's little theorem, a bit on quadratic residues.

And to top it all, I solved 2-3 hard problems of number theory from the IMO shortlists. These were N6 and N7 of the IMO 2003 shortlist as well as a problem from the IMO 2002 shortlist stating that the number of prime divisors of numbers of a certain form is at least 4^n. All of them boiled down to very similar manipulations of cyclotomic polynomials.

What I focused on was which part becomes obvious based on the general facts I have discussed and which parts involve the intuitive leap. And I ended with Pratt's result that primality testing is in nondeterministic polynomial time.

After lunch, I came back and discussed some growth functions in combinatorics, in particular, about Sidon sets and sum free sets. I threw in mention of greedy algorithms, methods involving two coordinates and two equations, and other stuff. I gave them the convolution free word problem which I had struggled with for a long time but which has a simple solution. Again... it'll take too much effort to write all those things down here, so either surf them on your own or wait for my articles to appear.

I moved on to infinitary combinatorics, discussing Konig's lemma and problems based on a similar trick: "any finite partition of an infinite set must contain at least one infinite part". And a related one: "any countable partition of an uncountable set must contain at least one uncountable part".

With that, my battery seemed to be going down, but phew... went on and discussed a few algebraic inequalities. For one of those inequalities (Problem 5 of IMO 2003), I learned of the rationale behind uncovering the solution from one of the team members, Riddhipratim Basu. I amplified his reasoning process, and related it to a lot of other things. Then I discussed an inequality from the IMO 2001 shortlist.

Finally, I did some side stuff in geometry. I began with lattice points and rational points, discussed a pretty problem (C5 of IMO 2003 shortlist) which was classified as combinatorics but was actually lattice geometry. I went ahead to discuss more on triangle centers, and asked many questions.

When I posed a problem, I didn't give them much time to think. Rather, I started bombarding them with ideas and other results. It's not that I believe people shouldn't think about problems, rather, I feel that people's initial thoughts should be colored with the richness of their past experiences in the subject. They need to keep it all right there and draw upon it even as they look at the problem. And that's the commitment that I wanted to convey.

As a past Olympiad goer, I have lost touch with Olympiad problems. It is possible that problems that I would have been able to solve earlier will now baffle me, and I might take more time over some problems. Nonetheless, there are some problem solving techniques I learnt there that have now become almost routine for me. So have certain ways of organizing knowledge and ideas. I know that if I want to pick up Olympiad mathematics again full steam, I can do a really great job of it... but of course i have other priorities right now.

But it is fun. Olympiad mathematics... and it was a defining experience for me.

A word to all here. Mathematics is a solitary activity... you have to live with and love your problems. But at the same time, it is a social activity. We can live and breathe and talk our problems, we can discuss then casually and seriously, we can share ideas in the subject. It is about depth and commitment. But our own Olympiad experience largely goes waste unless we share it both with our own endeavors in other fields and with other people who are starting out on the Olympiad track. So those of you who've been there or are going there... do blog it. Let's create a rich database of Olympiad experiences that will inspire further generations!

PS: If you are keen on learning where to find the problem details etc. Read my previous post (viz the one below this thanks to reverse chronological order) or directly visit my Olympiad resources page.