Problem 2
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: (1) no line passes through any point of the configuration; (2) 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.
Step 3 of 5: Pair the remaining points of the isolated colour
In plain words
Each same-colour pair can be enclosed in a thin strip bounded by two parallel lines.
Detailed analysis
After the first cut, there are 2012 red points left in the first case, or 2012 blue points left in the second. Pair these 2012 points arbitrarily. For each pair {A,B}, draw two lines parallel to AB and sufficiently close to AB that the closed strip between them contains no other configuration point. The pair is then isolated in a monochromatic region, and the two lines cost .