Two guards are overseeing the order in a room whose shape and dimensions are shown in the diagram. Each two adjacent walls are perpendicular to each other, and the dimensions are given in meters. The guards stand close to the wall at the locations marked with squares. Together, they can oversee the entire room, but only a part of the room can be overseen by both at the same time.

a) Mark the part of the room that both guards can oversee in the diagram.

b) How many times larger is the entire room compared to the part that both guards can oversee.

(K. Pazourek)

![](https://cdn.mathpix.com/cropped/2024_04_17_8b8f2fdcb46700a46ac3g-2.jpg?height=728&width=731&top_left_y=1806&top_left_x=662)
The entire room can be divided into squares with a side of 10 meters as shown in the image. The parts of the room visible to each guard are marked with two types of hatching. The part that both guards can see is thus doubly hatched:

![](https://cdn.mathpix.com/cropped/2024_04_17_8b8f2fdcb46700a46ac3g-3.jpg?height=574&width=580&top_left_y=495&top_left_x=744)

This part consists of 4 squares, while the entire room consists of 16 squares. Therefore, the entire room is four times larger than the part that both guards can see.

Evaluation. 2 points for correctly depicting the part visible to both guards; 2 points for the auxiliary division of the room; 2 points for solving the problem and providing the result.

Note. The entire room can be divided into coarser parts based on whether they are under the control of one or both guards. These are rectangles with dimensions $30 \times 20$, $20 \times 30$, and a square $20 \times 20$ (all in meters). The entire room corresponds to a rectangle with dimensions $80 \times 20$, and the doubly hatched part has dimensions $20 \times 20$, which is exactly one quarter of the whole.