Problem 8. Seryozha placed numbers from 1 to 8 in the circles so that each number, except one, was used exactly once. It turned out that the sums of the numbers on each of the five lines are equal. Which number did Seryozha not use?

![](https://cdn.mathpix.com/cropped/2024_05_06_03b105aa15b6031ee02fg-3.jpg?height=326&width=443&top_left_y=174&top_left_x=812)
Answer: 6.

Solution. Note that each number is contained in exactly two lines. Therefore, the doubled sum of all used numbers is five times greater than the sum of the numbers on one line. Thus, the sum of all numbers is divisible by 5.

Note that $1+2+3+4+5+6+7+8=36$. Therefore, for the sum of the used numbers to be divisible by 5, Seryozha must refrain from using the number one or six.

If Seryozha does not use the number one, then the sum of the numbers in each row will be equal to $(2+3+4+5+6+7+8) \cdot 2: 5=14$. But we have two rows, each containing two numbers, and 14 can be represented as the sum of two considered digits in only one way: $14=6+8$.

Thus, Seryozha does not use the digit 6. For example, he can accomplish the given task as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_03b105aa15b6031ee02fg-3.jpg?height=362&width=508&top_left_y=1144&top_left_x=777)