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 $I$.