1. A column of scouts has a length of $1 \mathrm{~km}$ and is moving uniformly. The courier, who is at the front of the column, runs to the end of the column, delivers the message, and then returns to the front of the column. During this time, the column travels a distance of $1 \mathrm{~km}$. How far did the courier travel?
Solution. Let the courier move with speed $x$, the column with speed $y$, and let $s$ denote the distance the courier travels from the front of the column to its end. Then we have:

$$
\frac{s}{x}=\frac{1-s}{y} ; \quad \frac{s+1}{x}=\frac{s}{y}
$$

From these two equations, it follows that

$$
\frac{x}{y}=\frac{s}{1-s}=\frac{s+1}{s} ;(s \neq 0, s \neq 1)
$$

from which we obtain:

$$
2 s^{2}=1 ; s=\frac{\sqrt{2}}{2}
$$

The distance traveled is

$$
2 s+1=2 \cdot \frac{\sqrt{2}}{2}+1=(\sqrt{2}+1) \text{ km}
$$