Question 153
Let $n \ge 4$. Let $M$ be a set of $n$ points in the plane, no three of which are collinear. Before the game starts, $n$ segments are drawn in the plane such that both endpoints of every segment belong to $M$ and every point of $M$ is an endpoint of exactly two segments.
We then consider the following operation: choose two segments $AB$ and $CD$ that have a common point (endpoints included), erase these two segments and draw $AC$ and $BD$.
Prove that it is impossible to perform $\frac{n^3}{4}$ or more operations.
We then consider the following operation: choose two segments $AB$ and $CD$ that have a common point (endpoints included), erase these two segments and draw $AC$ and $BD$.
Prove that it is impossible to perform $\frac{n^3}{4}$ or more operations.