Friday, 4 March 2016

An interesting application of the pigeonhole principle

Suppose 2016 points of the circumference of a circle are coloured red and the remaining points are coloured blue. Given any natural number $n \geq 3$, prove that there is a regular n-sided polygon all of whose vertices are blue. (Source: INMO 2016)

Consider a regular $(2017 \times n)$-gon on the circle, say with vertices $A_1, A_2, A_3,\dots, A_{2017 \times n}$. For each $j$, such that $1 \leq j \leq 2017$, consider the set of points $\{A_k : k \equiv j \mod 2017\}$. Note that these are the vertices of a regular $n$-gon, say $S_j$. Thus, we get 2017 regular $n$-gons: $S_1, S_2,\dots, S_{2017}$. Since there are only 2016 red points on the circumference, by pigeonhole principle there must be at least one $n$-gon among these 2017 which does not contain any red vertex. But then, all the vertices of this regular $n$-gon must be blue. Hence proved.

Thursday, 3 March 2016

Which one is greater - $e^{\pi}$ or $\pi^{e}$?

Recall that: $ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots $

Therefore, $ \forall x > 0 $, $ e^x > 1 + x $.

Substituting $x = \frac{\pi}{e} - 1$ in the above inequality, we get:

$ e^{\frac{\pi}{e} - 1} > \frac{\pi}{e}
 \implies e^{\frac{\pi}{e}} > \pi
 \implies e^{\pi} > \pi^{e} $

Tuesday, 1 March 2016


His mind was being invaded, and he was powerless to resist!
Where once resided only dry logic, rationality and offensive practicality,
A single innocent face started making its presence felt -
Never deserting his thoughts for even a single moment in the day.

Like a cancerous growth, the image kept spreading throughout his consciousness;
Day by day it expanded, taking up more and more space in his mind,
And pushing out every other thought till they were left tottering on the brink -
Dangling at the edge of the precipice by one last tenuous thread of reason.
Ultimately, it evolved into a burning passion that threatened to consume him -
Consume him and the entire universe of his existence in one raging inferno!

He was at a total loss to figure out what was happening to him;
The gears and cogs of his desperate mind whirred uselessly, and then fell silent;
But he was no closer to finding an explanation, and felt himself drowning in utter confusion!

Initially, he fought hard against this invasion -
Chided himself for this inexplicable madness,
Rebuked himself for this illogical obsession,
Scoffed at himself for chasing daydreams and fantasies!

Yet, little by little, he found himself unable to resist;
Till finally he was forced to abandon his ineffectual struggles and give in.
And then, he was surprised to find himself relishing this strange alien sensation,
Though he knew not what it was, nor could he find a name for it!

However, as She slowly poisoned the cold, terse rationalist to death,
And freed the poet long imprisoned in a deep, dark corner of his mind,
Realization dawned at last inside his half-paralyzed brain;
Realization of what he had become:
A hopeless addict, full of delusions!
And She? His intoxication!