13.266. Two athletes start running simultaneously - the first from $A$ to $B$, the second from $B$ to $A$. They run at different but constant speeds and meet at a distance of 300 m from $A$. After running the entire path $A B$, each immediately turns around and meets the other at a distance of 400 m from $B$. Find the length of $A B$.
Solution.

Consider the graphs of the runs of two athletes (Fig. 13.9). Let $AB = x$ m, $C$ and $D$ be the points of the first and second meetings, $v_{1}$ and $v_{2}$ be the speeds of the first and second athlete. Then $\frac{300}{v_{1}}=\frac{x-300}{v_{2}}\left({ }^{*}\right)$. During the time between the meetings $(KL)$, the athletes ran:

the first $(x-300)+400=x+100$ m;

the second $300+(x-400)=x-100$ m.

Thus, $\frac{x+100}{v_{1}}=\frac{x-100}{v_{2}}(* *)$.

From $\left({ }^{*}\right)$ and (**) we get $\frac{x+100}{300}=\frac{x-100}{x-300} \Rightarrow x=500 \mathrm{~m}$.

Answer: 500 m.