This solution was sent in by Harry 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:

$${x \over V_r} = {1 + x \over V_c}.$$

Similarly the time taken between the cyclist first passing the runner and the finish gives the second equation: \begin{equation}{1 - x\over V_r} = {x\over V_c}. \end{equation} The ratio of $V_c$ to $V_r$ from the two equations gives: \begin{equation*}{V_c\over V_r} = {1 + x\over x} = {x \over 1 - x} \end{equation*} Hence \begin{equation*}x^2 = 1 - x^2. \end{equation*} From this we get $x = \sqrt{1\over 2}$ and this gives the ratio of the speeds as \begin{equation*}{V_c\over V_r} = {{1 + 1/\sqrt 2}\over {1/\sqrt 2}} = \sqrt 2 + 1. \end{equation*}