# Difference between revisions of "2013 IMO Problems/Problem 2"

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A configuration of <math>4027</math> points in the plane is called ''Colombian'' if it consists of <math>2013</math> red points and <math>2014</math> blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is ''good'' for a Colombian | A configuration of <math>4027</math> points in the plane is called ''Colombian'' if it consists of <math>2013</math> red points and <math>2014</math> blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is ''good'' for a Colombian | ||

configuration if the following two conditions are satisfied: | configuration if the following two conditions are satisfied: | ||

− | + | *no line passes through any point of the configuration; | |

− | + | *no region contains points of both colours. | |

Find the least value of <math>k</math> such that for any Colombian configuration of <math>4027</math> points, there is a good | Find the least value of <math>k</math> such that for any Colombian configuration of <math>4027</math> points, there is a good | ||

arrangement of <math>k</math> lines. | arrangement of <math>k</math> lines. |

## Revision as of 00:44, 11 October 2013

## Problem

A configuration of points in the plane is called *Colombian* if it consists of red points and blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is *good* for a Colombian
configuration if the following two conditions are satisfied:

- no line passes through any point of the configuration;
- no region contains points of both colours.

Find the least value of such that for any Colombian configuration of points, there is a good arrangement of lines.

## Solution

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