You may also like

problem icon

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

problem icon

Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

problem icon

Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

Weekly Problem 13 - 2006

Stage: 3 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.

View the previous week's solution