Question 168
Ten children decide to run around a circular lake with a perimeter of $200~\mathrm{m}$. At the start of the race, the ten children are standing still, occupying distinct positions, and each of them will run clockwise or counterclockwise, depending on their wish, with speed $\frac{200}{k}~\mathrm{m} / \mathrm{min}$, where $k$ is a positive integer. Show that after some time, there will be an instant at which all the children will be exactly at their initial positions.