9.5. In a $5 \times 5$ table, all numbers from 1 to 25 are written, one number per cell. Consider all pairs of numbers that are in the same row or the same column. For each pair, the ratio of the larger number to the smaller number is calculated. The minimum of these ratios among all pairs is denoted by $A$. What is the maximum value that $A$ can take for different arrangements of the numbers in the table?
# Answer: $\frac{6}{5}$.

Solution. Consider the numbers $25,24,23,22,21$. If any two of them are in the same row or the same column, then $A \leq \frac{25}{21}\frac{6}{5}$. In the first column, the numbers differ by a multiple of 4; the smallest ratio is $\frac{21}{17}>\frac{6}{5}$. In the second column, the minimum ratio is $\frac{22}{18}>\frac{6}{5}$, in the third column $\frac{23}{19}>\frac{6}{5}$, in the fifth column $\frac{16}{12}>\frac{6}{5}$. In the fourth column, the minimum ratio is $\frac{24}{20}=\frac{6}{5}$. Therefore, $A=\frac{6}{5}$, and the estimate is achieved.

Comment. A fully justified solution - 7 points. Points obtained - 4 points, an example provided and justified - 3 points, points are cumulative. A correct example without justification - 2 points. No points are awarded for considering suboptimal examples. The problem is not solved or solved incorrectly - 0 points.