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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 line passes through any point of the configuration;
**no region contains points of both colours.
+
*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 $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ 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 $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.

Solution

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