All three runners finish at the same time.

Let the radius of $R$'s track be $r$ and let the radius of the first semicircle of $P$'s track be $p$; then the radius of the second circle of this track is $r-p$.

The total length of $P$'s track is $\pi p + \pi(r-p) = \pi r$, the same length as $R$'s track.

By a similar argument, the length of $Q$'s track is also $\pi r$.

*This problem is taken from the UKMT Mathematical Challenges.*