2014 IMO Problems/Problem 6

Problem

A set of lines in the plane is in $\textit{general position}$ 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 finite are; we call these its $\textit{finite regions}$ . 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.

Solution

Hi,

I was wondering if the problem seems a bit ambiguous. As there are n set of lines in consideration, with the constraint that no two are parallel and none of them are concurrent (constraints 1 and 2). Picking any three set of lines from set of n lines (nC3) will give a triangle formed by these lines because of the above constraints. This means if we color any more than 2 set of lines, at least 1 triangle is going to have all the three lines as blue. This sets the upper limit on the number of lines to be colored blue as 2 and not $\sqrt{n}$ . Is this right?

Thanks and regards

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

2014 IMO (Problems) • Resources
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2014 IMO Problems/Problem 6

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