## Category Archives: IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha)

### Games, social behavior, chess, economics and maths

The following is some trivia but in fact, not so trivia, in this age of data science, data analytics, social media platforms, on-line gaming etc…If you decide to ponder over deep…you will become a giant mathematician or applied mathematician or of course, a computer science wunderkind..

The following is “picked out as it is” from a famous biography, (which regular readers of my blog will now know, perhaps, is a favorite mathematical biography for me)…A Beautiful Mind by Sylvia Nasar, biography of mathematical genius, John Forbes Nash, Jr, Nobel Laureate (Economics) and Abel Laureate:

“It was the great Hungarian-born polymath John von Neumann who first recognized that social behaviour could be analyzed as games. Von Neumann’s 1928 article on parlor games was the first successful attempt to derive logical and mathematical rules about rivalries. Just as William Blake saw the universe in a grain of sand, great scientists have often looked for clues in vast and complex problems in the small, familiar phenomena of daily life. Isaac Newton reached insights about the heavens by juggling wooden balls. Albert Einstein contemplated a boat paddling upriver. Von Neumann pondered the game of poker.

A seemingly trivial and playful pursuit like poker, von Neumann argued, might hold the key to more serious human affairs for two reasons. Both poker and economic competition require a certain type of reasoning, namely the rational calculation of advantage and disadvantage based on some internally consistent system of values (“more is better than less’). And, in both, the outcome for any individual actor depends not only on his own actions, but on the independent actions of others.

More than a century earlier, the French economist Antoine-Augustin Cournot had pointed out that problems of economic cnoice were greatly simplified when either none or a large number of other agents were present. Alone on his island, Robinson Crusoe does not have to worry about whose actions might affect him. Neither do Adam Smith’s butchers and bakers. They live in a world with so many others that their actions, in effect, cancel each other out. But when there is more than one agent but not so many that their influence may be safely ignored, strategic behavior raises a seemingly insoluble problem:”I think that he thinks that I think that he thingks,” and so forth…

So play games but think math ! 🙂

Nalin Pithwa

### You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…

https://www.cs.virginia.edu/~robins/YouAndYourResearch.html

### A magic trick!

You may have heard of a magic trick that goes like this. Take any number. Add 5. Double the result. Subtract 6. Divide by 2. Subtract 2. Now tell me your answer, and I will tell you what you have started with.

Pick a number and try it.

You can see what is going on if you let x be your original number, and follow the steps to make a formula $f(x)$ for the number you end up with.

Have fun !!!

Shared by Nalin Pithwa,

Ref: Calculus and Analytic Geometry by G B Thomas, 9th edition.

### Theory of Equations: III: IITJEE maths: algebra

may overlap a bit with previous lecture(s)…

### Applications of Derivatives: IITJEE Maths tutorial problem set: III

Slightly difficult questions, I hope, but will certainly re-inforce core concepts:

1. Prove that the segment of the tangent to the curve $y=c/x$ which is contained between the co-ordinate axes, is bisected at the point of tangency.
2. Find all tangents to the curve $y=\cos{(x+y)}$ for $-\pi \leq x \leq \pi$ that are parallel to the line $x+2y=0$.
3. Prove that the curves $y=f(x)$, where $f(x)>0$ and $y=f(x)\sin(x)$, where $f(x)$ is a differentiable function, have common tangents at common points.
4. Find the condition that the lines $x\cos{\alpha} + y \sin{\alpha}=p$ may touch the curve $(\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1$.
5. If $p_{1}$ and $p_{2}$ are lengths of the perpendiculars from origin on the tangent and normal to the curve $x^{2/3} + y^{2/3}=a^{2/3}$ respectively, prove that $4p_{1}^{2}+p_{2}^{2}=a^{2}$.
6. Show that the curve $x=1-3t^{2}$, $y=t-3t^{3}$ is symmetrical about x-axis and has no real points for $x>1$. If the tangent at the point t is inclined at an angle $\psi$ to OX, prove that $3t = \tan{\psi} + \sec{\psi}$. If the tangent at $P(-2,2)$ meets the curve again at Q, prove that the tangents at P and Q are at right angles.
7. A tangent at a point $P_{1}$ other than $(0,0)$ on the curve $y=x^{3}$ meets the curve again at $P_{2}$. The tangent at $P_{2}$ meets the curve at $P_{3}$ and so on. Show that the abscissae of $P_{1}, P_{2}, \ldots, P_{n}$ form a GP. Also, find the ratio of area $\frac{\Delta P_{1}P_{2}P_{3}}{area \hspace{0.1in} P_{2}P_{3}P_{4}}$.
8. Show that the square roots of two successive natural numbers greater than $N^{2}$ differ by less than $\frac{1}{2N}$.
9. Show that the derivative of the function $f(x) = x \sin {(\frac{\pi}{x})}$, when $x>0$, and $f(x)=0$ when $x=0$ vanishes on an infinite set of points of the interval $(0,1)$.
10. Prove that $\frac{x}{(1+x)} < \log {(1+x)} < x$ for $x>0$.

More later, cheers,

Nalin Pithwa.

### How to solve equations: Dr. Vicky Neale: useful for Pre-RMO or even RMO training

Dr. Neale simply beautifully nudges, gently encourages mathematics olympiad students to learn to think further on their own…

### Paul Erdos, Mathematics, Russia and USA:

It is true …universally, including India…

Hats off to the “Intellect of the Wise Mathematicians”, late, adorable professor of mathematics, Paul Erdos.

— humble tribute …from Nalin Pithwa.

Paul Erdos, most profound quote ever made by any mathematician