This solution was sent in by Harry Li, age 15, from Riccarton High School, Christchurch, New Zealand

A cyclist and a runner are pracising on a race track going round at constant speeds, $V_{c}$ for the cyclist and $V_{r}$ for the runner. Let the fraction of the circuit covered by the runner when they first meet be $x$ and then the cyclist will have covered $1 + x$ circuits. Equating the time taken gives the first equation:

$\frac{x}{V_{r}} = \frac{1 + x}{V_{c}} .$ $(1)$
Similarly the time taken between the cyclist first passing the runner and the finish gives the second equation:

$\frac{1 - x}{V_{r}} = \frac{x}{V_{c}} .$ $(2)$
The ratio of $V_{c}$ to $V_{r}$ from the two equations gives:

$\frac{V_{c}}{V_{r}} = \frac{1 + x}{x} = \frac{x}{1 - x}$

Hence

$x^{2} = 1 - x^{2} .$

From this we get $x = \sqrt{\frac{1}{2}}$ and this gives the ratio of the speeds as

$\frac{V_{c}}{V_{r}} = \frac{1 + 1 / \sqrt{2}}{1 / \sqrt{2}} = \sqrt{2} + 1 .$