2023 APMO 第 5 题

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2 n-1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows:
First, he chooses an endpoint of each segment as a "sink". Then he places the present at the endpoint of the segment he is at. The present moves as follows:

- If it is on a line segment, it moves towards the sink.
- When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink.

If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove Tony can send presents to exactly $n$ of his $2 n-1$ friends.

平面上有$n$条线段，没有三条线段相交于一点，并且每对线段在各自的内部相交一次。托尼和他的 $2 n-1$ 朋友各自站在一条线段的不同端点上。托尼希望给他的每一位朋友送圣诞礼物如下：

首先，他选择每个线段的一个端点作为“汇点”。然后他将礼物放在他所在段的端点处。目前的动向如下：

- 如果它在线段上，则它向水槽移动。
- 当它到达两个线段的交点时，它会改变其行进的线段并开始向新的汇点移动。

如果礼物到达端点，则该端点上的朋友可以收到他们的礼物。证明托尼可以向他的 $2 n-1$ 朋友中的 $n$ 发送礼物。