Problem 1. Two runners started simultaneously in the same direction from one point of a circular distance, and the first runner, having moved ahead, caught up with the second again at the moment when the second had only run half a circle. From this moment, the second runner doubled his speed. Will the first runner catch up with him now? And if he does, how many laps will the second runner have completed by that time?
Solution. From the first condition of the problem, it follows that the speed of the first runner is 3 times greater than the speed of the second (since he ran $3 / 2$ laps, while the second ran $1 / 2$ lap), that is, $V_{1}=3 V_{2}$. Therefore, even after doubling the speed of the second runner, the first will still move faster. So the answer to the first question is: yes.

Let the second runner run $k$ laps between the first and second overtaking. Then

$$
k S=2 V_{2} \cdot t, \quad(k+1) S=V_{1} \cdot t
$$

From this, $\frac{k+1}{k}=\frac{V_{1}}{2 V_{2}}=\frac{3}{2} \Rightarrow k=2$.

Answer: Will overtake. The second runner will run 2.5 laps from the start of the movement.