Problem 2. In a small town, there is only one tram line. It is circular, and trams run along it in both directions. There are stops on the loop: Circus, Park, and Zoo. The journey from Park to Zoo via Circus is three times longer than not via Circus. The journey from Circus to Zoo via Park is twice as short as not via Park. Which route from Park to Circus is shorter - via Zoo or not via Zoo - and by how many times?

(A.V. Shapovalov)
Answer: The path not through the Zoo is 11 times shorter.

Solution. Let's board the tram at the Zoo stop and travel through the Circus to the Park, and then, without leaving the tram, return to the Zoo. The second part of the journey is three times shorter than the first, meaning the first part takes up three quarters of the full circle, and the second part takes up one quarter. Let's mark the Zoo and the Park on the map, and somewhere on the longer arc between them, mark the Circus (see the diagram). Now, on the same tram, let's travel from the Circus to the Zoo (passing the Park, as shown on the map).

![](https://cdn.mathpix.com/cropped/2024_05_06_79cfe44052259f2639fcg-04.jpg?height=357&width=392&top_left_y=302&top_left_x=581)

Having arrived at the Zoo, we will return to the Circus on the same tram, completing a full circle. The first part of the journey is twice as short as the second, meaning it takes up one third of the circle. This means that the path (still on the same tram) from the Circus to the Park will not pass through the Zoo and will constitute $\frac{1}{3}-\frac{1}{4}=\frac{1}{12}$ of the full circle. The path through the Zoo is $1-\frac{1}{12}=\frac{11}{12}$ of the circle, which is 11 times longer.