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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 $\pi r$.
Running Race
Age 14 to 16
ShortChallenge Level 
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.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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