18. Lucky Spy (from 9th grade, 2 points). A spy with a radio transmitter parachutes into a completely random point in a perfectly square forest measuring $10 \times 10$ km. Upon landing, he immediately starts a radio transmission, revealing important information to the enemy.

At each of the four corners of the forest, there are radio direction finders, each capable of detecting the transmitter's operation within a 10 km radius. To accurately determine the coordinates of the operating transmitter, the transmitter must be detected by at least two direction finders.

The spy is lucky: one of the radio direction finders is not working today. What is the probability that the counterintelligence will not be able to determine the coordinates of the location where the spy landed?
Solution. Suppose, for definiteness, that the direction finder in the northeast corner is not working (Fig. 4). The diagram shows how many direction finders "see" each of the six resulting regions. The location of the spy cannot be accurately determined if he is in one of the two regions visible to only one direction finder.

We will assume that the side of the square is equal to 1. We introduce the notation as shown in Fig. 5, and find the area of the shaded figure, consisting of an equilateral triangle $A E D$ and two identical sectors $A B E$ and $D E C$.

The area of the triangle is $\frac{\sqrt{3}}{4}$, and each sector has a central angle of $30^{\circ}$, so the area of each sector is one-twelfth of the area of a unit circle, which is $\frac{\pi}{12}$. Therefore, the area of the entire shaded figure is $\frac{\sqrt{3}}{4}+\frac{\pi}{6}$, and the area of the curvilinear triangle $B E C$ is $1-\frac{\sqrt{3}}{4}-\frac{\pi}{6}$. The area of the entire zone where the spy's location cannot be determined is twice as large: $2-\frac{\sqrt{3}}{2}-\frac{\pi}{3} \approx 0.087$.

Answer: approximately $0.087$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2370d2b995b387125216g-39.jpg?height=559&width=569&top_left_y=222&top_left_x=1361)

Fig. 4.

![](https://cdn.mathpix.com/cropped/2024_05_06_2370d2b995b387125216g-39.jpg?height=556&width=593&top_left_y=828&top_left_x=1357)

Fig. 5.