2017 IMO Shortlist C5

A hunter and an invisible rabbit play a game in the Euclidean plane. The hunter's starting point $H_{0}$ coincides with the rabbit's starting point $R_{0}$. In the $n^{\text {th }}$ round of the game $(n \geq 1)$, the following happens. (1) First the invisible rabbit moves secretly and unobserved from its current point $R_{n-1}$ to some new point $R_{n}$ with $R_{n-1} R_{n}=1$. (2) The hunter has a tracking device (e.g. dog) that returns an approximate position $R_{n}^{\prime}$ of the rabbit, so that $R_{n} R_{n}^{\prime} \leq 1$. (3) The hunter then visibly moves from point $H_{n-1}$ to a new point $H_{n}$ with $H_{n-1} H_{n}=1$. Is there a strategy for the hunter that guarantees that after $10^{9}$ such rounds the distance between the hunter and the rabbit is below 100 ?

猎人和隐形兔子在欧几里得平面上玩游戏。猎人的起点$H_{0}$与兔子的起点$R_{0}$重合。在游戏 $(n \geq 1)$ 的第 $n^{\text {th }}$ 轮中，会发生以下情况。 (1) 首先，隐形兔子从当前点 $R_{n-1}$ 秘密且不被观察地移动到某个新点 $R_{n}$，其中 $R_{n-1} R_{n}=1$。 (2) 猎人有一个跟踪设备(例如狗)，它返回兔子的大概位置 $R_{n}^{\prime}$，使得 $R_{n} R_{n}^{\prime} \leq 1$。 (3) 然后猎人明显地从点 $H_{n-1}$ 移动到新点 $H_{n}$，其中 $H_{n-1} H_{n}=1$。猎人是否有一种策略可以保证在 $10^{9}$ 这样的回合之后猎人和兔子之间的距离低于 100 ？