2016 IMO 第 6 题

There are $n\geq 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n - 1$ times. Every time he claps, each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.

(a) Prove that Geoff can always fulfill his wish if $n$ is odd.

(b) Prove that Geoff can never fulfill his wish if $n$ is even.

平面中有 $n\geq 2$ 条线段，每两条线段相交，且没有三条线段交于一点。杰夫必须选择每个线段的一个端点，并在其上面向另一个端点放置一只青蛙。然后他会拍手 $n - 1$ 次。每次他拍手，每只青蛙都会立即向前跳到其线段上的下一个交点。青蛙永远不会改变跳跃的方向。杰夫希望以这样的方式放置青蛙，使得它们中的任何两个都不会同时占据同一个交叉点。

(a) 证明如果 $n$ 是奇数，Geoff 总能实现他的愿望。

(b) 证明如果 $n$ 是偶数，Geoff 永远无法实现他的愿望。