Difference between revisions of "2014 IMO Problems/Problem 6"
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==Problem== | ==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 <math>n</math>, in any set of <math>n</math> lines in general position it is possible to colour at least <math>\sqrt{n}</math> of the lines blue in such a way that none of its finite regions has a completely blue boundary. | + | A set of lines in the plane is in <math>\textit{general position}</math> 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 <math>\textit{finite regions}</math>. Prove that for all sufficiently large <math>n</math>, in any set of <math>n</math> lines in general position it is possible to colour at least <math>\sqrt{n}</math> of the lines blue in such a way that none of its finite regions has a completely blue boundary. |
==Solution== | ==Solution== | ||
Revision as of 05:41, 9 October 2014
Problem
A set of lines in the plane is in 
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 
. Prove that for all sufficiently large 
, in any set of lines in general position it is possible to colour at least 
of the lines blue in such a way that none of its finite regions has a completely blue boundary.
Solution
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 2014 IMO (Problems) • Resources | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
