# What is the kind of concentration required for RMO INMO IMO and IITJEE Advanced Mathematics?

Laser like concentration is a pre-requisite to genuinely creative work in math, computer science, or any other field, like chess and music. Let me illustrate a story of John Nash Jr. (Nobel Laureate, Economics, Abel Laureate genius mathematician): (Reference: A Beautiful Mind by Sylvia Nasar, Chapter 5 Genius):

KAI LAI CHUNG, a mathematics instructor who had survived the horrors of the Japanese conquest of his native China, was surprised to see the door of the Professors’ Room standing ajar.  It was usually locked. Kai Lai liked to stop by on the rare occasions when it was open and nobody was about. It had the feel of an empty church, no longer imposing and intimidating as it was in the afternoons when it was crowded with mathematical luminaries, but simply a beautiful sanctuary.

The light in the west common room filtered through thick stained-glass windows inlaid with formulae: Newton’s law of gravity, Einstein’s theory of relativity, Heisenberg’s uncertainty principle of quantum mechanics. At the far end, like an altar, was a massive stone fireplace. On one side was a carving of a fly confronting
the paradox of the MObius band. MObius had given a strip of paper a half twist and connected the ends, creating a seemingly impossible object: a surface with only one side. Kai Lai especially liked to read the whimsical inscription over the fire-place, Einstein’s expression of faith in science, “Der Herr Gott ist raffiniert aber Boshaft ist Er nicht,” which he took to mean that “the Lord is subtle but not malicious.”

On this particular fall morning, as he reached the threshold of the half-open door, Kai Lai stopped abruptly. A few feet away, on the massive table that dominated the room, floating among a sea of papers, sprawled a beautiful dark-haired young man. He lay on his back staring up at the ceiling as if he were outside on a lawn under an elm looking up at the sky through the leaves, perfectly relaxed, motionless, obviously lost in thought, arms folded behind his head. He was whistling softly. Kai Lai recognized the distinctive profile immediately. It was the new graduate student from West Virginia. A trifle shocked and a little embarrassed,
Kai Lai backed away from the door and hurried away before Nash could see or hear him.

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Cheers,
Nalin Pithwa

# Pre RMO algebra : some tough problems

Question 1:

Find the cube root of $x^{3} -12x^{2} + 54x -112 + \frac{108}{x} - \frac{48}{x^{2}} + \frac{8}{x^{3}}$

Question 2:

Find the square root of $\frac{x}{y} + \frac{y}{x} +3 - 2\sqrt{\frac{x}{y}} -2\sqrt{\frac{y}{x}}$

Question 3:

Simplify (a):

$(\frac{x}{x-1} - \frac{1}{x+1}). \frac{x^{3}-1}{x^{6}+1}.\frac{(x-1)^{2}(x+1)^{2}+x^{2}}{x^{4}+x^{2}+1}$

Simplify (b):
$\{ \frac{a^{4}-y^{4}}{a^{2}-2ay+y^{2}} \div \frac{a^{2}+ay}{a-y} \} \times \{ \frac{a^{5}-a^{3}y^{2}}{a^{3}+y^{3}} \div \frac{a^{4}-2a^{3}y+a^{2}y^{2}}{a^{2}-ay+y^{2}}\}$

Question 4:

Solve : $\frac{3x}{11} + \frac{25}{x+4} = \frac{1}{3} (x+5)$

Question 5:

Solve the following simultaneous equations:

$2x^{2}-3y^{2}=23$ and $2xy - 3y^{2}=3$

Question 6:

Simplify (a):

$\frac{1- \frac{a^{2}}{(x+a)^{2}}}{(x+a)(x-a)} \div \frac{x(x+2a)}{(x^{2}-a^{2})(x+a)^{2}}$

Simplify (b):

$\frac{6x^{2}y^{2}}{m+n} \div \{\frac{3(m-n)x}{7(r+s)} \div \{ \frac{4(r-s)}{21xy^{2}} \div \frac{(r^{2}-s^{2})}{4(m^{2}-n^{2})}\} \}$

Question 7:

Find the HCF and LCM of the following algebraic expressions:

$20x^{4}+x^{2}-1$ and $25x^{4}+5x^{3} - x - 1$ and $25x^{4} -10x^{2} +1$

Question 8:

Simplify the following using two different approaches:

$\frac{5}{6- \frac{5}{6- \frac{5}{6-x}}} = x$

Question 9:

Solve the following simultaneous equations:

Slatex x^{2}y^{2} + 192 = 28xy\$ and $x+y=8$

Question 10:

If a, b, c are in HP, then show that

$(\frac{3}{a} + \frac{3}{b} - \frac{2}{c})(\frac{3}{c} + \frac{3}{b} - \frac{2}{a})+ \frac{9}{b^{2}}=\frac{25}{ac}$

Question 11:

if $a+b+c+d=2s$, prove that

$4(ab+cd)^{2} - (a^{2}+b^{2}-c^{2}-d^{2})^{2}= 16(s-a)(s-b)(s-c)(s-d)$

Question 12:

Determine the ratio $x:y:z$ if we know that

$\frac{x+z}{y} = \frac{z}{x} = \frac{x}{z-y}$

More later,
Nalin Pithwa

Those interested in such mathematical olympiads should refer to:

https://olympiads.hbcse.tifr.res.in

(I am a tutor for such mathematical olympiads).

# Elementary Number Theory, ISBN numbers and mathematics olympiads

Question 1:

The International Standard Book Number (ISBN) used in many libraries consists of nine digits $a_{1} a_{2}\ldots a_{9}$ followed by a tenth check digit $a_{10}$ (somewhat like Hamming codes), which satisfies

$a_{10} = \sum_{k=1}^{9}k a_{k} \pmod {11}$

Determine whether each of the ISBN’s below is correct.
(a) 0-07-232569-0 (USA)
(b) 91-7643-497-5 (Sweden)
(c) 1-56947-303-10 (UK)

Question 2:

When printing the ISBN $a_{1}a_{2}\ldots a_{9}$, two unequal digits were transposed. Show that the check digits detected this error.

Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.

More later,
Nalin Pithwa.

# Mathematics Olympiads: A curious calculation and its cute proof !!

Explain why the following calculations hold:

$1.9 + 2 =11$
$12.9 + 3 = 111$
$123.9 + 4 = 1111$
$1234.9 + 5 = 11111$
$12345.9 + 6 = 111111$
$123456.9 + 7 = 1111111$
$1234567.9 + 8 = 11111111$
$12345678.9 + 9 = 111111111$
$123456789.9 + 10 = 11111 11111$

Hint:

Show that $(10^{n-1}+2.10^{n-2}+3.10^{n-3}+ \ldots + n)(10-1) + (n+1)=\frac{10^{n+1}-1}{9}$

More later,
Nalin Pithwa

# A good way to start mathematical studies …

I would strongly suggest to read the book “Men of Mathematics” by E. T. Bell.

It helps if you start at a young age. It doesn’t matter if you start later because time is relative!! 🙂

Well, I would recommend you start tinkering with mathematics by playing with nuggets of number theory, and later delving into number theory. An accessible way for anyone is “A Friendly Introduction to Number Theory” by Joseph H. Silverman. It includes some programming exercises also, which is sheer fun.

One of the other ways I motivate myself is to find out biographical or autobiographical sketches of mathematicians, including number theorists, of course. In this, the internet is an extremely useful information tool for anyone willing to learn…

Below is a list of some famous number theorists, and then there is a list of perhaps, not so famous number theorists — go ahead, use the internet and find out more about number theory, history of number theory, the tools and techniques of number theory, the personalities of number theorists, etc. Become a self-learner, self-propeller…if you develop a sharp focus, you can perhaps even learn from MIT OpenCourseWare, Department of Mathematics.

Famous Number Theorists (just my opinion);

1) Pythagoras
2) Euclid
3) Diophantus
4) Eratosthenes
5) P. L. Tchebycheff (also written as Chebychev or Chebyshev).
6) Leonhard Euler
7) Christian Goldbach
8) Lejeune Dirichlet
9) Pierre de Fermat
10) Carl Friedrich Gauss
11) R. D. Carmichael
12) Edward Waring
13) John Wilson
14) Joseph Louis Lagrange
15) Legendre
16) J. J. Sylvester
11) Leonoardo of Pisa aka Fibonacci.
15) Srinivasa Ramanujan
16) Godfrey H. Hardy
17) Leonard E. Dickson
18) Paul Erdos
19) Sir Andrew Wiles
20) George Polya
21) Sophie Germain
24) Niels Henrik Abel
25) Richard Dedekind
26) David Hilbert
27) Carl Jacobi
28) Leopold Kronecker
29) Marin Mersenne
30) Hermann Minkowski
31) Bernhard Riemann

Perhaps, not-so-famous number theorists (just my opinion):
1) Joseph Bertrand
2) Regiomontanus
3) K. Bogart
4) Richard Brualdi
5) V. Chvatal
6) J. Conway
7) R. P. Dilworth
8) Martin Gardner
9) R. Graham
10) M. Hall
11) Krishnaswami Alladi
12) F. Harary
13) P. Hilton
14) A. J. Hoffman
15) V. Klee
16) D. Kleiman
17) Donald Knuth
18) E. Lawler
19) A. Ralston
20) F. Roberts
21) Gian Carlo-Rota
22) Bruce Berndt
23) Richard Stanley
24) Alan Tucker
25) Enrico Bombieri

Happy discoveries lie on this journey…
-Nalin Pithwa.