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:

x
V_r
= 1 + x
V_c
.
(1)
Similarly the time taken between the cyclist first passing the runner and the finish gives the second equation:

1 − x
V_r
= x
V_c
.
(2)
The ratio of V_c to V_r from the two equations gives:

V_c
V_r
= 1 + x
x
= x
1 − x

Hence

x² = 1 − x².

From this we get
x =
√

¹/₂

and this gives the ratio of the speeds as

V_c
V_r
= 1 + 1/√2
1/√2
= √2 + 1.