Pre RMO Training: Plane geometry with combinatorics

Question 1:

There are 4 possible ways to place three distinct lines in a plane. Two of these configurations involve parallel lines, the other two do not. Draw all these possibilities including the one which encloses a region.

Question 2:

Prove that the sum of any two sides of a triangle is greater than the third side. Hint: Use the following permissible clever argument: the shortest distance joining any two distinct points is given by a straight line joining those two points.

Question 3:

There are 8 possible ways to place 4 distinct lines in a plane. Five of these configurations involve parallel lines; the other three do not. Draw all the possibilities.

Remark: Questions like 1 and 2 are at the heart of combinatorics questions in plane geometry in pre RMO and RMO.

Nalin Pithwa

PS: Prove the parallelogram law: |a+b| \leq |a|+|b|

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.


Nalin Pithwa

Prof. Tim Gowers’ on recognising countable sets

Thanks Dr. Gowers’. These are invaluable insights into basics. Thanks for giving so much of your time.

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:

(I am a tutor for such mathematical olympiads).