How did I prepare?

By the time I entered eleventh standard, I had a lucid career goal: I wanted to do research in mathematics. However, despite being "good" at school mathematics, I was not sure of my standing in the subject. While CMI and ISI were definitely the places I wanted to go, I still wanted to prepare for and give the IIT JEE so that, if all else failed, I could do a mathematics course at one of the IITs.

Preparation for the IIT JEE in India is highly competitive, and most "serious" students enroll in a regular coaching class to prepare. I joined Vidyamandir Classes (VMC), one of Delhi's most reputed coaching institutes. Their regular lectures began in early May. Thus, after lazing around and reading mathematics here and there in the month of April, I was forced to "buckle my shoes" in May.

This is not the place to describe what happened in the VMC lectures. However, I'll still say a little bit on how I found them and how they affected my overall preparation plans. The first class at VMC was a physics lecture, and I was pretty impressed by the thoroughness with which the physics lecturer covered the topics. The second class was in mathematics, and the part of mathematics covered was both new and interesting. I was understandably quite taken in with the mathematics person as well. The third class was in chemistry. The lecturer taught chemistry for only 5-10 minutes,and spent the rest of the time giving general philosophical advice. Although I felt this was overkill on advice, I enjoyed hearing some of the pieces of advice.

However, what disillusioned me with Vidyamandir was not just the teachers but the whole attitude and atmosphere. If I were to describe it in one word: it would be "scared". Students seemed under a lot of "pressure" to perform and do well in the IIT, in order to get a career. I felt myself a misfit because : (i) I wasn't all that interested in engineering -- I wanted to do math (ii) I didn't feel that kind of pressure either from within and without.

Within my peer group in VMC, I was lonely in my deep interest and passion for mathematics. Under the guise of sharpening thinking skills and of doing problems relevant for IIT, I was able to discuss some Olympiad related ideas with my peer group in VMC. Nonetheless, there was nobody with whom to discuss my overall Olympiad preparation plans and strategies.

During the months of May-June, apart from the formality of attending the VMC lectures and doing some of their problems, I focussed on Olympiad mathematics. As I already mentioned, I had neither any Olympiad senior nor any peer to discuss my progress with, so I had to find my own way. I'll describe some specific instances.

The methodology of research has been classified into theory building and problem solving. By nature, I'm more of a theory builder: during my tenth class, I used to make notes to try to fit a pattern into what was being taught in school, and in junior classes, I had tried to create new axiomatic theories of geometry. When I wanted to master a particular area, i wouldn't try to solve a hundred problems in it, rather, I would solve just a few and try to stare at the linking ideas of the theory.

Olympiad problems, on the other hand, are famous for the lack of pattern and its intractability even with knowledge of the theory. But I wanted to have a theory-centered approach to the preparation, in addition to practising lots of problems. There were two reasons. One, I felt (correctly) that today's insights are tomorrow's theories. Two, I believed that since my ultimate aim is to study mathematics, knowing the theory will be helpful in later study.

Now for my area wise progress in the four Olympiad areas during the months of May-June:

(I) Geometry: At the time, I had only Challenge and Thrill of Pre-College Mathematics and Engel's book to prepare from for geometry.

From Challenge and Thrill, I picked up Ceva's and Menelaus' Theorem, and did a lot of practice questions on these results. There were some questions, specially in the last exercise, that I just couldn't solve at the time. However, coming back to them in a few days, I was able to crack them, and the solutions remained etched in my mind. I converted all the difficult problems into important theorems, and framed them in my mind.

I also read Penguin's Dictionary of Curious and Interesting Geometry. I would open a random page from the Penguin's Dictionary, see a definition or a result, and try to prove it. Many of the results which seemed simple to state were hard to prove, and I got stuck over a number of them. Again, it was a matter of getting them eventually after a long struggle.

On the theory side, I was keen to build a reasonable "theory of quadrilaterals". Quadrilaterals are a lot more messy than triangles, and there are all sorts of special kinds of quadrilaterals. After analyzing the many special kinds of quadrilaterals, I came up with the following scheme.

Let a,b,c,d be the sides of the quadrilateral. Consider properties:

s1: a + c = b + d
s2: a - c = b - d, s2': a - c = d - b
s3: ac = bd
s4: a/c = b/d, s4': a/c = d/b
s5: a^2+c^2= b^2 + d^2
s6: a^2-c^2= b^2 - d^2, s6': a^2 - c^2 = d^2 - b^2

Let P, Q, R, S be the angles. Consider properties:

a1: P + R = Q + S
a2: P - R = Q - S, a2': P - R = S - Q

Let p be the length of the segment from the vertex P to the point of intersection of diagonals. Analogously, define q,r,s.

d1: p + r = q + s
d2: p - r = q - s, d2': p - r = s - q
d3: pr = qs
d4: p/r = q/s d4': p/r = s/q

Now, we have the following:
(i) s1 is precisely the property of being a "circumscribed quadrilateral", a quadrilateral having an incircle.
(ii) s3 is precisely the property of being a "proportional quadrilateral", a quadrilateral where the angle bisectors at two opposite vertices meet in the diagonal joining the other two.
(iii) s5 is the property of being a "right quadrilateral", a quadrilateral where the diagonals are perpendicular.
(iv) If any two of s1,s3, s5 hold then so does the third. In that case, a = b and c = d
or a = d and b = c. Thus, the quadrilateral is a "kite".
(v) If both s2 and s2' (resp. s4 and s4', resp s6 and s6') hold, then a =c and b = d. Thus, the quadrilateral is a "parallelogram".
(vi) a1 is precisely the property of being a "cyclic quadrilateral"
(vii) a2 and a2' together force a parallelogram.
(viii) If P = Q and R = S, then the quadrilateral is a "trapezium"
(ix) d1 is the property of being an "equidiagonal" quadrilateral.
(x) d2 and d2' (resp. d4 and d4') collectively imply the property of being a parallelogram.
(xi) d3 is the property of being a cyclic quadrilateral.
(xii) d1 and d3 together imply the property of being a parallelogram.

Properties that require only "one" constraint: circumscribed, proportional, right, cyclic, equidiagonal

Properties that require "two" constraints: kite, parallelogram, trapezium

Further, we have:

kite and parallelogram = rhombus (3 constraints)
trapezium and parallelogram = rectangle (3 constraints)
all three = square (4 constraints)

When I look back on it, even four years later, I cannot help marvelling at the beauty of what I had observed. And I feel that this is an indication that I have research potential.

(II) Combinatorics: I started off with the Schaum's Outline Series in Combinatorics. This was a completely problem-oriented book, but each problem contained a new morsel of the theory. I worked on the first chapter, trying each problem, looking at the solution, and augmenting my theoretical understanding. I also created heuristics lists. Although I probably never used those heuristics lists explicitly, the process of creating the heuristics itself helped me clarify my thinking.

I also attempted the puzzles in Mathematical Circles. These taught me some elementary combinatorial tricks and also made for good puzzles to discuss with others.

(III) Number Theory: I started on Burton in earnest. The simple and elegant proof of Fermat's Little Theorem was one factor that hooked me to number theory early on. I finished the first 5-6 chapters of Burton's books, getting stuck at the quadratic residues part. I was to later return to quadratic residues with a vengeance.

The clear notation and notion of congruences helped me give neat solutions to problems where I had hitherto given ad hoc solutions.

I also had the book by Niven and Zuckermann, but I did not read it during the summer holidays.

(IV) Algebra: I started reading the parts on "Inequalities" and "Functional equations" in Engel's book. At the time, I was not too interested in algebra per se, though I liked to use algebraic methods and tools in general. Besides, the VMC maths classes already had a reasonable amount of algebra in them.

On the whole, the months of May and June were crucial in kickstarting me into the world of Olympiad mathematics.

The beginning

My tryst with mathematics began in Class II when I learnt the "test for divisibility by 9". Between Class III and IV, I learnt algebra (translate: the use of letters for unknowns). Right uptil class ninth, I stayed a class or two ahead of school.

I also exposed myself to some hobby math and popular math books. These included Keith Devlin's book on "Mathematics: The New Golden Age", which had big stuff on prime numbers and the Riemann hypothesis and P versus NP. I didn't follow all the details at the time, but did get intrigued by a result called Fermat's little theorem. Devlin's book stated that this was a piece of elementary but ingenuous mathematics, and I always believed that any "elementary problem" could be solved with enough time and dedication. So I took the problem with me to the dentist's appointment but came back empty handed.

The problem was -- I had nothing to chew upon. With no approach, no idea of how to start, no knowledge of what can be used, I was completely clueless. All I knew was that x - 1 divided xⁿ - 1, and that is not sufficient in itself.

I was always keen on mathematics. During my seventh and eighth standards, I set myself the less taxing goal of being a math school teacher. I even used to rehearse in my mind a scene where I was teaching a class. I visualized how I would arrange and organize the curriculum. I thought of setting up my own school. Interestingly, when I was in seventh standard, I dreamed of being a seventh standard teacher, and when I reached eighth standard I dreamed of being an eighth standard teacher. You can imagine what happened when I reached ninth standard.

By the time I reached tenth standard, I realized that my dream of being a teacher and elucidator of the subject could be best met if I took up the line of research. Then, I could teach in colleges and spread the word for mathematics in schools. Which meant that I should take up "mathematical research". But what did that mean? Would I become a musty old professor in a college? Or would I be a cutting edge thinker, an explorer into the world of knowledge?

Another question that plagued me was: where to study? Should I take up an undergraduate course in mathematics? What "career options" did such a course offer? How severely limiting was it compared to the much sought engineering degree. What was the best place to do mathematics.

The first piece of knowledge was from a friend of a friend of my mother. He was the National Coordinator for the Olympiads at that time, and he worked in ISI Delhi at the time. He said that the two best places for B.Sc. in mathematics were Chennai Mathematical Institute and ISI Bangalore. He recommended these over the IITs for mathematics. Among these, he said that the CMI course was more demanding and better.

The next input that validated these two names came from some high up chap in government. I was a topper in one of the National Science Olympiads (this is a private competition and has nothing to do with the international olympiads) and attended a dinner with the Secretary for Science and Technology of the time. He again recommended the same names: ISI and CMI.

I also learnt that students who got through the INMO and attended the IMOTC twice (as Junior and Senior) got direct admission into the B.Sc. Math programme at CMI (Also see page on CMI at Wikipedia). This incerased my motivation to study for the Olympiads.

Towards the end of my tenth standard, I bought the book Challenge and Thrill of Pre College Mathematics written by the people in the Mathematical Olympiad Cell of India. Due to the tenth class CBSE examinations, I postponed reading the book. After my tenth class, I also requested an uncle to get a book from abroad: Problem Solving Strategies by Arthur Engel. I had come across this book by searching on Amazon.

I also went to the homepage of the Delhi Mathematical Olympiad Association. There, I discovered the names of a few more good books on the Olympiads. One of the books was available with the Olympiad coordinator in Delhi, Dr. Amitabh Tripathi, who works in IIT Delhi. Some time in April (after my board examinations) I went to visit Dr. Tripathi to buy the book. I asked him about options for studying mathematics, and he said that within India, CMI and ISI are the best places for doing mathematics, though he felt that the curriculum at both places was advanced and should be improved.

Another book I tried ordering from an Indian retailer was: Mathematical Olympiad Challenges by Titu Andreescu and Razvan Gelca. This book took a really long time to come, but, as I'll describe later, it was a real gem once it came.

Well, this was the beginning...

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.

About this blog

It feels good to blog about an area of activity I've been passionate about for a long time: the Mathematics Olympiad. First, a quick look at my achievements in the Olympiads: I've represented India at the International Mathematical Olympiads in the years 2003 at Japan and 2004 at Greece. I won silver medals both times, being the top scorer from the Indian team.

Now, if you've come to this blog to collect serious information about the IMOs and the training process in India, I must warn you that this is more in the tune of my personal chronicles. Of course, it may contain a lot of information and a lot of ideas and I hope it inspires more students! But for official information, check out the Wikipedia entries on the IMO, the IMO Training Camp, and the INMO. These have all the relevant links.

For specific suggestions and names of books and websites, check out my page on Olympiad Resources which has an article on Preparing for Olympiads.

But do come back and read this blog as well! It discusses things like:

(i) How I prepared for the Olympiads: the usual issues like theory versus practice, does it conflict with JEE preparation etc. are bound to surface.
(ii) What role the Olympiads had in my decision to pursue mathematics: I am doing B.Sc. (Hons) in Mathematics and Computer Science at the Chennai Mathematical Institute. But may be that's getting a little ahead of the story!
(iii) What problem solving really means and how it relates to research. You might also like to see my What Is Research? blog in this regard.
(iv) The perils and the pains, the joys and the mundanity, of solving Olympiad problems.

So for whom is this intended? For anybody who is:

(i) Keen on Olympiads, or on preparing for them
(ii) Eager to revive his or her own memories of the wonderful days
(iii) Keen to learn about how a life in mathematics might begin

See you soon with more details!