2017 IMO 第 3 题

A hunter and an invisible rabbit play a game in the plane. The rabbit and hunter start at points $A_{0} = B_{0}$ . In the $n$ th round of the game ( $n \geq 1$ ), three things occur in order:

(i) The rabbit moves invisibly from $A_{n - 1}$ to a point $A_{n}$ such that $A_{n - 1}A_{n} = 1$ .

(ii) The hunter has a tracking device (e.g. dog) which reports an approximate location $P_{n}$ of the rabbit, such that $P_{n}A_{n} \leq 1$ .

(iii) The hunter moves visibly from $B_{n - 1}$ to a point $B_{n}$ such that $B_{n - 1}B_{n} = 1$ .

Let $N = 10^{9}$ . Can the hunter guarantee that $A_{N}B_{N} < 100$ ?

猎人和隐形兔子在飞机上玩游戏。兔子和猎人从点 $A_{0} = B_{0}$ 开始。在游戏的第 $n$ 轮中 ( $n \geq 1$ )，三件事按顺序发生：

(i) 兔子无形地从 $A_{n - 1}$ 移动到点 $A_{n}$，使得 $A_{n - 1}A_{n} = 1$ 。

(ii) 猎人有一个跟踪设备(例如狗)，它报告兔子的大致位置 $P_{n}$，使得 $P_{n}A_{n} \leq 1$ 。

(iii) 猎人明显地从 $B_{n - 1}$ 移动到点 $B_{n}$，使得 $B_{n - 1}B_{n} = 1$ 。

让 $N = 10^{9}$ 。猎人能保证 $A_{N}B_{N} < 100$ 吗？