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.