Question 7
In a coordinate city there are $n \ge 3$ tramlines parallel to the $x$-axis such that each line begins from $x$-coordinate $0$ and ends at $x$-coordinate $n$. Exactly one tram of length $1$ is moving on each line: on the first line with speed $1$, on the second line with speed $2$ etc, until on the last line with the speed $n$. When a tram reaches the end of its line it instantly starts moving back without turning around. In the morning all trams start moving at the same time from the starting position where the $x$-coordinate of the back end of the tram is $0$. Prove that the trams' projections onto the $x$-axis never cover the whole interval from $0$ to $n$. (Grade 12.)