ISL 2006 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)