2006 IMO Shortlist C1

We have $n \geq 2$ lamps $L_{1}, \ldots, L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: - if the lamp $L_{i}$ and its neighbours (only one neighbour for $i=1$ or $i=n$, two neighbours for other $i$ ) are in the same state, then $L_{i}$ is switched off; - otherwise, $L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. (a) Prove that there are infinitely many integers $n$ for which all the lamps will eventually be off. (b) Prove that there are infinitely many integers $n$ for which the lamps will never be all off. (France)

我们有 $n \geq 2$ 灯 $L_{1}, \ldots, L_{n}$ 连续，每个灯要么打开，要么关闭。每秒我们同时修改每盏灯的状态，如下所示： - 如果灯 $L_{i}$ 及其邻居($i=1$ 或 $i=n$ 仅有一个邻居，其他 $i$ 为两个邻居)处于相同状态，则 $L_{i}$ 关闭； - 否则，$L_{i}$ 打开。最初，除了最左边的一盏灯亮起外，所有灯都关闭。 (a) 证明有无数个整数$n$，最终所有的灯都会熄灭。 (b) 证明有无限多个整数 $n$ 使得灯永远不会全部熄灭。 (法国)