2014 IMO Shortlist C7

Let $M$ be a set of $n \geq 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $A B$ and $C D$ sharing a common interior point and replace them by the segments $A C$ and $B D$ if none of them is present at this moment. Prove that it is impossible to perform $n^{3} / 4$ or more such moves. (Russia)

令 $M$ 为平面上的一组 $n \geq 4$ 点，其中没有三个点共线。最初，这些点与 $n$ 线段连接，以便 $M$ 中的每个点恰好是两个线段的端点。然后，在每一步中，可以选择共享公共内点的两个线段$A B$和$C D$，并且如果此时它们都不存在，则用线段$A C$和$B D$替换它们。证明不可能执行 $n^{3} / 4$ 或更多这样的动作。 (俄罗斯)