Problem 10.2. On a sheet of paper, three intersecting circles are drawn, forming 7 regions. We will call two regions adjacent if they share a common boundary. Regions that border at exactly one point are not considered adjacent.

Two regions already have numbers written in them. Write integers in the remaining 5 regions so that the number in each region equals the sum of all numbers in adjacent regions. What number should be in place of the question mark?

![](https://cdn.mathpix.com/cropped/2024_05_06_62ad5020cf8c9b2474f9g-2.jpg?height=395&width=420&top_left_y=89&top_left_x=517)
Answer: -8.

Solution. Let the unknown number be $x$. Then we can place the numbers in the other regions as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_62ad5020cf8c9b2474f9g-2.jpg?height=390&width=412&top_left_y=657&top_left_x=525)

The sum of the numbers in the regions surrounding $x$ should be equal to $x$. We can write this as:

$$
(x+15)+(x+8)+(x-7)=x
$$

Solving the equation, we get $x=-8$. It is easy to verify that the condition of the problem is also satisfied for the other regions.