Yashchenko I.V.

On a circular road, there are four gas stations: $A, B, C$, and $D$. The distance between $A$ and $B$ is 50 km, between $A$ and $C$ is 40 km, between $C$ and $D$ is 25 km, and between $D$ and $A$ is 35 km (all distances are measured along the circular road in the shortest direction).

a) Provide an example of the arrangement of the gas stations (with the distances between them) that satisfies the condition of the problem.

b) Find the distance between $B$ and $C$ (list all possibilities).
First, determine the arrangement of gas stations $A, C$, and $D$.

## Solution

The problem provides all three distances between $A, C$, and $D$. First, let's determine the arrangement of these three gas stations. Gas stations $A$ and $C$ divide the circular road into two arcs. If gas station $D$ were on the shorter arc, the sum of the distances from $A$ to $D$ and from $D$ to $C$ would be equal to the distance from $A$ to $C$. However, this is not the case.

Therefore, gas station $D$ must be on the longer arc, so the length of the longer arc between $A$ and $C$ is $A D + D C = 25 + 35 = 60$ km. Consequently, the length of the circular road is 60 km $+ A C = 100$ km.

Since $B A = 50$ km, $A$ and $B$ are diametrically opposite. Therefore, the distance from $B$ to $C$ is $50 - 40 = 10$ km (see the diagram).

![](https://cdn.mathpix.com/cropped/2024_05_06_a1c7cb3d3682c5b3cf5fg-15.jpg?height=403&width=326&top_left_y=2491&top_left_x=869)

## Answer

a) See the diagram; b) 10 km.

![](https://cdn.mathpix.com/cropped/2024_05_06_a1c7cb3d3682c5b3cf5fg-16.jpg?height=412&width=334&top_left_y=134&top_left_x=862)