Can the circles on a five-pointed star (see figure) be filled with 4 ones, 3 twos, and 3 threes such that the sums of the four numbers on each of the five lines are equal?

![](https://cdn.mathpix.com/cropped/2024_05_06_ef1f6e5b136aa760f4dfg-07.jpg?height=203&width=211&top_left_y=402&top_left_x=932)

#
Suppose the numbers have been arranged in the required manner. Let $S$ be the sum of the numbers on each line, then the sum of the numbers on all five lines is $5S$. Since each circle lies at the intersection of two lines, in this calculation, the number written in each circle is counted twice. Therefore, the found sum is equal to twice the sum of all the placed numbers, that is, $5S = 2(4 \cdot 1 + 3 \cdot 2 + 3 \cdot 3)$. We get that $5S = 38$, which is impossible, since four integers stand on each line and their sum must be an integer.

## Answer

no, it is not possible.