2009 APMO 第 5 题

Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a $90^{\circ}$ left turn after every $\ell$ kilometer driving from start; Rob makes a $90^{\circ}$ right turn after every $r$ kilometer driving from start, where $\ell$ and $r$ are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction.

Let the car start from Argovia facing towards Zillis. For which choices of the pair $(\ell, r)$ is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

拉里和罗布是两个机器人，乘坐一辆车从阿尔戈维亚前往齐利斯。两个机器人都根据以下算法控制转向和转向：拉里从起点开始每行驶 $\ell$ 公里后左转 $90^{\circ}$ ；从起点开始每行驶 $r$ 公里后，Rob 就会右转 $90^{\circ}$，其中 $\ell$ 和 $r$ 是相对素的正整数。如果两个转弯同时发生，汽车将继续行驶而不改变方向。假设地面平坦，汽车可以向任意方向移动。

让汽车从阿尔戈维亚出发，面向齐利斯。对于 $(\ell, r)$ 对中的哪一个选择，汽车可以保证到达 Zillis，无论距离 Argovia 有多远？