Difference between revisions of "2013 IMO Problems/Problem 2"
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==Problem== | ==Problem== | ||
| − | A | + | 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: | |
| − | + | **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 | + | 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 01:43, 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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