6.1. On a circular road, there are three cities: $A, B$ and $C$. It is known that the path from $A$ to $C$ is 3 times longer along the arc not containing $B$ than through $B$. The path from $B$ to $C$ is 4 times shorter along the arc not containing $A$ than through $A$. How many times shorter is the path from $A$ to $B$ along the arc not containing $C$ than through $C$?
Answer: 19

Solution. From the condition of the problem, the following relationship follows:

$$
\begin{aligned}
& x=3(y+z) \\
& 4 z=x+y \\
& 3 y+3 z=4 z-y \\
& 4 y=z \\
& x=15 y \\
& x+z=19 y
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_01174ccc998f2eafc50eg-07.jpg?height=557&width=531&top_left_y=641&top_left_x=748)