Transcendental Lonely Runner

Question

Four runners start at the same position on a circular track which is 1 km long, and all run in the same direction around and around the track. The runners’ speeds are √​5​​, e, 3, and pi, π all in m/s. What is the first time (measured in seconds after the start) that the speed-e runner is at a distance of at least 250 m from every other runner?

Answer 1

We may look at the track from the point of view of the speed-e runner. Thus we consider the speed-e runner to remain at position $1$ on the unit circle in the complex plane, while at time $t$ (measured in units of $1000$ seconds) the other three are at $\exp(2\pi i v_j t)$ where $v_j = \sqrt{5}-e$, $3-e$ and $\pi - e$. We want $\cos(2 \pi v_j t) \le 0$ for $j = 1, 2, 3$. Plotting these cosines,

we see that the first time is when $2\pi (3-e) t = \pi/2$, at $t = 1/(12-4e)$.