Answer: $\large\frac{3v_1v_2v_3}{v_2v_3+v_1v_3 + v_1v_2}$

In general terms, the time taken to cover a third of the distance $x$ with speed $v$ is $t = \large\frac{\text{x}/3}{\text{v}}$

$\Rightarrow$ the overall speed $ \overline{v} = \Large\frac{x}{ \frac{x}{3v_1} + \frac{x}{3v_2} + \frac{x}{3v_3}}$

$\Rightarrow \overline{v} = \large\frac{3v_1v_2v_3}{v_2v_3+v_1v_3 + v_1v_2}$