### Rotating Triangle

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

### 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?

### 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:

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
View the current weekly problem