# Runner’s High (Speed)

I find the following mind-boggling.

Suppose that runner $$R_0$$ runs distance $$[0,d_0]$$ with average speed $$v_0$$. Runner $$R$$ runs $$[0,d]$$ with $$d>d_0$$ and with average speed $$v > v_0$$. I would have thought that by some application of the intermediate value theorem we can find a subinterval $$I\subseteq [0,d]$$ having length $$d_0$$ such that $$R$$ had average speed at least $$v_0$$ on $$I$$. This is not necessarily so!

Question. What is the smallest value of $$C\in\mathbb{R}$$ with $$C>1$$ and the following property?

Whenever $$d>d_0$$, and $$R$$ runs $$[0,d]$$ with average speed $$Cv_0$$, then we can guarantee that there is a subinterval $$I\subseteq [0,d]$$ having length $$d_0$$ such that $$R$$ had average speed at least $$v_0$$ on $$I$$.