2016 IMO Shortlist C7

Let $n \geq 2$ be an integer. In the plane, there are $n$ segments given in such a way that any two segments have an intersection point in the interior, and no three segments intersect at a single point. Jeff places a snail at one of the endpoints of each of the segments and claps his hands $n-1$ times. Each time when he claps his hands, all the snails move along their own segments and stay at the next intersection points until the next clap. Since there are $n-1$ intersection points on each segment, all snails will reach the furthest intersection points from their starting points after $n-1$ claps. (a) Prove that if $n$ is odd then Jeff can always place the snails so that no two of them ever occupy the same intersection point. (b) Prove that if $n$ is even then there must be a moment when some two snails occupy the same intersection point no matter how Jeff places the snails.

设 $n \geq 2$ 为整数。在平面中，有 $n$ 个线段，任何两个线段在内部都有交点，并且没有三个线段相交于一个点。杰夫将一只蜗牛放在每个线段的端点之一，然后拍手 $n-1$ 次。每次他拍手时，所有蜗牛都会沿着自己的线段移动，并停留在下一个交点，直到下一次拍手。由于每条线段上有 $n-1$ 个交点，因此所有蜗牛在 $n-1$ 拍手后都会到达距离起点最远的交点。 (a) 证明如果 $n$ 是奇数，那么 Jeff 总是可以放置蜗牛，使得它们中的任何两个都不会占据相同的交点。 (b) 证明如果 $n$ 是偶数，那么无论 Jeff 如何放置蜗牛，都一定存在两个蜗牛占据相同交点的时刻。