13.390 A column of cars moving uniformly at the same speed has a length of $5 \mathrm{km}$. The head of the column is in the last car, and a motorcyclist is next to him. On the order of the head, the motorcyclist increased his speed, caught up with the lead car, delivered a package, instantly turned around, and returned to his place at the same speed he had while moving forward. While the motorcyclist was completing the task, the column moved forward by 5 km. How many kilometers did the motorcyclist travel?
Solution. Let $x$ be the speed of the motorcyclist, $y$ be the speed of the column. Then $\frac{5}{x-y}$ is the time it takes for the motorcyclist to travel from the end of the column to its beginning, $\frac{5}{x+y}$ is the time it takes for the motorcyclist to travel from the beginning of the column to its end, and $\frac{5}{y}$ is the time it takes for the column to move while the motorcyclist is performing the task. Therefore,

$\cdot \frac{5}{x-y}+\frac{5}{x+y}=\frac{5}{y} \Rightarrow x^{2}-2 x y-y^{2}=0 \Rightarrow \frac{x}{y}=1+\sqrt{2}$.

Thus, the motorcyclist traveled a total of $\frac{5}{y} \cdot x$ km.

Answer: $5(1+\sqrt{2})$ km.