3. Is it possible to arrange 4 ones, 3 twos, and 3 threes in the circles of a five-pointed star (see figure) so that the sums of the four numbers on each of the five lines are equal?

![](https://cdn.mathpix.com/cropped/2024_05_06_4df8f76a1e5c1d2910b3g-2.jpg?height=200&width=202&top_left_y=865&top_left_x=1007)
# Solution

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 $5 S$. 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 numbers placed, that is, $5 S=2(4 \cdot 1+3 \cdot 2+3 \cdot 3)$. We get that $5 S$ $=38$, which is impossible, since on each line there are 4 integers and their sum must be an integer.

Answer: No, it is not possible.