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Running Race

Stage: 3 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Three race tracks made from semi-circles

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.
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