Two athletes are running on concentric circular tracks of radii R...

Since both athletes complete one full lap in the same time, their time periods $T$ are equal.

Angular speed is given by $\omega = \dfrac{2\pi}{T}$. Since $T$ is the same, $\omega$ is the same for both athletes.

The linear speed is related to angular speed by $v = \omega r$.

Therefore, the ratio of their linear speeds is $\dfrac{v_{1}}{v_{2}} = \dfrac{\omega R}{\omega (2R)} = \dfrac{1}{2}$, which is $1:2$.

Answer: $1:2$