How to use this blog?

Hi, welcome to my blog. I started this blog to share some of my small discoveries as a high school math-enthusiast. Over time it evolved into a massive repository of notes, problem collections, and other resources to help students prepare for mathematical olympiads and other contests, entrance exams of ISI, CMI, and more.

Please browse through the menu, and take a tour of the blog to locate materials that pertain to your interest. I hope you will find this blog helpful.

P.S. Sorry for the ads. To keep the blog freely accessible to anyone with the internet, and not having the money to fund it myself, I could not upgrade the website.

On teaching complex numbers

Complex numbers are usually introduced to high-school students in a rather confusing manner. Till a certain grade they are told that the square-root of a negative real number is “forbidden”, and then is introduced all of a sudden. It is a pity that despite being able to solve a ton of problems, most students cannot perceive complex numbers beyond mere algebraic manipulations. The true potential of complex numbers is perhaps better understood through a geometric viewpoint, followed by its applications in other topics such as geometry, trigonometry, polynomials, combinatorics, and so on.

Telescoping sums and products

Expressing sums or products as ‘telescoping’ ones is a very standard trick which I have seen to be quite enjoyed by high school students, although it is hardly introduced at the ordinary high school level (at least in India). Despite its simplicity it is widely applicable and used regularly in the academia.

Let me illustrate the main idea via the following simple examples:

Question. What is the value of the sum

What’s the chance that a randomly picked pen will be an unused one?

When I was in junior school, there was a very popular (and cheap) pen named agni icy gel pen. It is probably still in production, but the popularity has somewhat reduced. These pens came with transparent caps and there used to be another tiny cap on the pen tips, which we used to remove and throw away before using the pen for the first time. So if this tiny cap is present, that implies the pen has not been used yet.

Recently one of my friends, Rajarshi, asked me this intriguing question: suppose a school student buys some of these pens (say n many) before his final exams. In every exam he randomly picks one of these pens, removes the tiny cap if the pen is new, and starts writing. Now on any particular day (say on the k-th day), some of his pens are already used and some are new/unused. If he picks one of these pens randomly, what is the probability that he will pick an unused pen?

A Probabilistic Proof of Stirling’s Formula

What is Stirling’s formula?

It is familiar that the quantity grows very quickly with . Stirling’s formula gives an approximate value of which can be computed more easily and is quite accurate even for small values of . The formula states that

where denotes that

Here I am going to discuss a probabilistic proof of Stirling’s formula. The proof is originally due to Rasul A. Khan [2].

A Game of Chance

Suppose that we have a game of chance with the following rules:

1. In each round, a fair coin is tossed twice.
2. If it is HH, you get $30. If it is TT, you lose $10.
3. If you get one H and one T, then you have an option to toss the coin once more.
   • If you decide to toss the coin once more, you gain or lose $20 according to as it results in H or T.
   • If you decide not to toss the coin for the third time, then you lose $10.

Each of Alice and Bob plays the above game. Alice does not like to take risks, so she always decides not to toss the coin for the third time (if the first two tosses result in HT or TH). However, Bob likes to take risks and hence he always decides to toss the coin for the third time (if the first two tosses result in HT or TH). Can you tell who will gain more in the long run, Alice or Bob?

An Equilateral Triangle Associated with Fermat Point

Abstract: Suppose that P is the Fermat point of triangle ABC. We show that the nine-point centres of the triangles APB, BPC and CPA form an equilateral triangle. Furthermore, if G be the centroid of ABC, then PG is a diameter of the circumcircle of that equilateral triangle.

An Identity Involving Bernoulli Numbers and a Proof of Euler’s Formula for Zeta(2n)

Abstract: Let be the sequence of Bernoulli numbers, with In this note, we derive an identity involving Bernoulli numbers. Using that, we give a proof of Euler’s formula for

Sample Questions for Interview at ISI

Following is a collection of problems, for students who will give an interview at ISI for B.Stat/B.Math. Please note that these are not taken from the actual interviews. As per my understanding, I have tried to compile some problems which are at the same level.

Sample Interview Problems

For other details/tips for the interview, please visit Tips for Interview of ISI Entrance.

Tips for Interview of ISI Entrance

Some students who will be facing an interview for ISI Entrance have asked me to provide some details/tips for the interview. So here it is:

The first thing to do is: Don’t Worry. It is probably your first interview, and trust me, the interviewers know that it is your first time. Hence they are really very friendly. And since you came up to this point, they already know that you are better than many others. So there is no reason to worry much about the interview.