13.279. Two checkpoints divide a ski trail into three segments of equal length. It is known that the skier covered the path consisting of the first and second segments together at an average speed of $a$ m/min; the path consisting of the second and third segments together, he covered at an average speed of $b$ m/min. The average speed of the skier on the second segment was the same as the average speed for the first and third segments together. What is the average speed of the skier on the entire trail and on each segment of this trail separately? Conduct an analysis of the conditions for the existence of a real solution to the problem.
Solution.

Let the length of one section of the track be 1. Let $t_{1}, t_{2}, t_{3}$ be the time it takes for the skier to pass each section of the track. According to the problem,

$$
\left\{\begin{array}{l}
\frac{2}{t_{1}+t_{2}}=a, \\
\frac{2}{t_{2}+t_{3}}=b, \\
\frac{1}{t_{2}}=\frac{2}{t_{1}+t_{3}},
\end{array} \text { from which } t_{1}=\frac{3 b-a}{2 a b}, t_{2}=\frac{a+b}{2 a b}, t_{3}=\frac{3 a-b}{2 a b} .\right.
$$

The average speed over the entire track will be $\frac{3}{t_{1}+t_{2}+t_{3}}=\frac{2 a b}{a+b}$; on the first section $-\frac{1}{t_{1}}=\frac{2 a b}{3 b-a}$; on the second $-\frac{1}{t_{2}}=\frac{2 a b}{a+b}$; on the third $-\frac{1}{t_{3}}=\frac{2 a b}{3 a-b}$. The problem has a real solution if $\left\{\begin{array}{l}3 a-b>0, \\ 3 b-a>0,\end{array}\right.$, from which $\frac{b}{3}<a<3 b$. Answer: $\frac{2 a b}{a+b} ; \frac{2 a b}{3 b-a} ; \frac{2 a b}{a+b} ; \frac{2 a b}{3 a-b}$ m/min, where $\frac{b}{3}<a<3 b$.