Problem 5.3. The figure shows a plan of the road system of a certain city. In this city, there are 8 straight streets, and 11 intersections are named with Latin letters $A, B, C, \ldots, J, K$.

Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets. Which three intersections should the police officers be placed at? It is sufficient to provide at least one suitable arrangement.

All streets run along straight lines.

Horizontal streets: $A-B-C-D, E-F-G, H-I-J-K$.

Vertical streets: $A-E-H, B-F-I, D-G-J$.

Diagonal streets: $H-F-C, C-G-K$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0323bbf84409a1adeb34g-09.jpg?height=359&width=601&top_left_y=886&top_left_x=420)
Answer: $B, G, H$.

Solution. The option will work if police officers are placed at intersections $B, G, H$. It can be shown that this is the only possible option.

Since there are only three vertical streets, and each must have one police officer, there are definitely no police officers at intersections $C$ and $K$. Therefore, there must be a police officer at intersection $G$. Then, on street $E-F-G$, there are no more police officers (since there is at least one police officer on streets $A-B-C-D$ and $H-I-J-K$). On street $C-F-H$, there must be a police officer, and he can only be placed at intersection $H$. The streets $A-B-C-D$ and $B-F-I$ remain uncovered, so the third police officer needs to be placed at intersection $B$.