Question 90
Let $M$ be convex 2011-gon. Consider 2011 points lying inside $M$ and such that no three of all 4022 points (the vertexes of $M$ and 2011 points inside $M$) are collinear. A coloring of all points in two colors is called good if it is possible to connect some of the points by segments such that the following conditions hold:
(1) Each segment has its endpoints of one and the same color.
(2) No segments intersect in inner points.
(3) Between any two points of the same color there exists a path through the given segments.
Find the number of all good colorings.
(1) Each segment has its endpoints of one and the same color.
(2) No segments intersect in inner points.
(3) Between any two points of the same color there exists a path through the given segments.
Find the number of all good colorings.