13.143. A completes a certain work in a time that is $a$ days longer than $B$, and $b$ days longer than $C$. Working together, $A$ and $B$ complete this work in the same number of days as $C$. Determine the time it takes for each to complete the work individually. Under what relationship between the given quantities does the problem have a solution?
## Solution.

Let $A$ complete the work in $x$ days, $B$ in $x-a$ days, and $C$ in $x-b$ days. According to the condition, $\frac{1}{x}+\frac{1}{x-a}=\frac{1}{x-b}$, from which $x=b+\sqrt{b(b-a)}$ days; for $b>a$, $x-a=b-a+\sqrt{b(b-a)}$ days; $x-b=\sqrt{b(b-a)}$ days.

Answer: $\quad b+\sqrt{b(b-a)} ; b-a+\sqrt{b(b-a)} ; \sqrt{b(b-a)}$ days; the problem has a solution when $b>a$.