19. In a $3 \times 3$ square, the cells were filled with numbers $1,2,3, \ldots, 9$. Then, in each circle (Fig. 2), the arithmetic mean of the four surrounding numbers was written. After that, the arithmetic mean of the four obtained numbers was calculated. What is the largest number that can result from this?
19. The largest number will be obtained in the case where the circles contain the largest possible arithmetic means. When determining these arithmetic means, the numbers in the corner cells are taken once, the number in the central cell is taken four times, and the numbers in the middle cells adjacent to the sides of the square are taken twice. Based on this, we will write the largest number 9 in the central cell, and the smallest numbers $-1, 2, 3,$ and 4 in the corner cells (Fig. 13). As a result, we get $x_{1}=\frac{21}{4} ; x_{2}=6 ; x_{3}=\frac{27}{4} ; x_{4}=\frac{26}{4}$ and $\left(x_{1}+x_{2}+x_{3}+x_{4}\right): 4=(21+24+27+26): 16=\frac{98}{16}=6.125$.

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-171.jpg?height=522&width=529&top_left_y=276&top_left_x=1186)

Fig. 13