Shapovalov A.V.

Cities $A, B$, and $C$ together with the straight roads connecting them form a triangle. It is known that the direct route from $A$ to $B$ is 200 km shorter than the detour through $C$, and the direct route from $A$ to $C$ is 300 km shorter than the detour through $B$. Find the distance between cities $B$ and $C$.
The route $B-A-C$ (from $B$ to $C$ via $A$) is 500 km shorter than the route $B-C-A-B-C$: the length of segment $B A$ is 200 km less than the length of the route $B-C-A$, and the length of segment $A C$ is 300 km less than the length of the route $A-B-C$. The second route differs from the first by two segments $B C$. Therefore, $B C=500: 2=250$ km.

## Answer

250 km.