IMO Shortlist 2014 problem C7
Let be a set of points in the plane, no three of which are collienar. Initially these points are connected with segments so that each point in is the endpoint of exactly two segments. Then, at each step, one may choose two segments and sharing a common interior point and replace them by the segments and if none of them is present at this moment. Prove that it is impossible to perform or more such moves.