Answer
There is a time during the race when the two runners have the same speed.
Work Step by Step
Let $t_1$ be the time when the race started.
Let $t_2$ be the time when the race ended.
Let $g(t)$ be the position function of one runner.
Let $h(t)$ be the position function of the other runner.
Let $f(t) = g(t) - h(t)$
We can assume that $g(t)$ and $h(t)$ are continuous on the time interval $[t_1, t_2]$, and differentiable on the time interval $(t_1, t_2)$. Then $f(t)$ is continuous on the time interval $[t_1, t_2]$, and differentiable on the time interval $(t_1, t_2)$.
According to the Mean Value Theorem, there is a number $c$ in the interval $(t_1,t_2)$ such that $f'(c) = \frac{f(t_2)-f(t_1)}{t_2-t_1}$
Then:
$f'(c) = \frac{f(t_2)-f(t_1)}{t_2-t_1}$
$f'(c) = \frac{[g(t_2)-h(t_2)]-[g(t_1)-h(t_1)]}{t_2-t_1}$
$f'(c) = \frac{(0)-(0)}{t_2-t_1}$
$f'(c) = 0$
$g'(c)-h'(c) = 0$
$g'(c) = h'(c)$
Since $g'(t)$ and $h'(t)$ are the velocity functions of the two runners, there is a time during the race when the two runners have the same velocity. Since speed is the absolute value of velocity, there is a time during the race when the two runners have the same speed.
