The number of good colourings in 2011-gon
Source: Bulgaria MO 2011
May 30, 2011
inductioncombinatorial geometrycombinatorics proposedcombinatorics
Problem Statement
In the interior of the convex 2011-gon are points, such that no three among the given points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied:
1) Each segment joins two points of the same colour.
2) None of the line segments intersect.
3) For any two points of the same colour there exists a path of segments connecting them.
Find the number of "good" colourings.