Problem 5. In each circle of the diagram, one of the numbers $1,2,3,4$ and 5 must be written, so that the sum of the numbers in the three vertical circles is equal to the sum of the numbers in the three horizontal circles. How many solutions does the problem have?
Solution. From the set $\{1,2,3,4,5\}$, two subsets need to be chosen such that they have an empty intersection, and the sums of the elements in each of them are equal. This can be done in the following ways:

![](https://cdn.mathpix.com/cropped/2024_06_05_1cb7beb611b796adc950g-02.jpg?height=295&width=1079&top_left_y=893&top_left_x=323)

$1^{\circ}\{3,4\}$ and $\{2,5\}$

$2^{\circ}\{1,5\}$ and $\{2,4\}$

$3^{\circ}\{4,1\}$ and $\{3,2\}$

In the case of $1^{\circ}$, we have the sum $3+4=2+5$, in the case of $2^{\circ}$, we have $1+5=2+4$, and in the third case, we have $4+1=3+2$. Therefore, the problem has three solutions.

## V section - nine-year-old