Question 37
There are $n \ge 2$ line segments in the plane such that every two segments cross at a point interior to both, and no three segments have a common point. Mafalda must choose one of the endpoints of each segment and place a frog on it, facing the other endpoint. Then she will whistle $n - 1$ times. At each whistle, every frog immediately jumps forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Mafalda wants to place the frogs in such a way that no two of them ever occupy the same intersection point at the same time.
1. Prove that if $n$ is odd, Mafalda can always achieve her goal.
2. Prove that if $n$ is even, Mafalda can never achieve her goal.
1. Prove that if $n$ is odd, Mafalda can always achieve her goal.
2. Prove that if $n$ is even, Mafalda can never achieve her goal.