IMO 2014/6

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finte area; we call these its finite regions. Prove that for all sufficiently large $n$ , in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ of the lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with $\sqrt{n}$ replaced by $c \sqrt{n}$ will be awarded points depending on the value of the constant $c$ .

#Geometry #Combinatorics #IMO

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Does anyone have a solution for this one? Based on what I've heard from the Indian IMOers, only one of them attempted it (and got 2). I'll work on this in school.

Sreejato Bhattacharya - 6 years, 11 months ago

Only 15 people got full marks for this. I'm not too sure how to proceed with this, and I'm guessing that a probabilistic expectation method is used.

Calvin Lin Staff - 6 years, 11 months ago

I've heard that there exists a purely elementary solution: one can simply construct an algorithm which satisfies the problem conditions. However since this is an IMO P6 and I extremely suck at combinatorics, I haven't made much significant progress yet.

And well, a probabilistic approach given by Evan Chen (TWN 2; v_Enhance on AoPS) gives a bound of $\dfrac{1}{\sqrt[3]{6e}}\sqrt{n}$ which is unfortunately smaller than $\sqrt{n}.$ There should exist a solution using the probabilistic method, though. Also, I have heard that the bound can actually be improved to $\sqrt{n \ln (n)}.$

Sreejato Bhattacharya - 6 years, 11 months ago

There exists an elementary solution using the extremal principle. Surprisingly, the probabilistic method isn't needed for c = 1.

Zi Song Yeoh - 6 years, 11 months ago

Note: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $n$ .

That single sentence got me scared at the IMO. I didn't even read the problem after that. :)

Shenal Kotuwewatta - 6 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$