8.1. In a rectangle of size $5 \times 18$, numbers from 1 to 90 were placed. This resulted in five rows and eighteen columns. In each column, the median number was selected, and from these median numbers, the largest one was chosen. What is the smallest value that this largest number can take?

Recall that among 99 numbers, the median is the one that is greater than 49 others and less than 49 others.
Answer: 54

Solution. The largest of the average numbers $N$ is not less than each of the eighteen selected averages. Each of the eighteen averages, in turn, is not less than three numbers from its column (including itself). In total, $N$ is not less than some 54 numbers in the table. Therefore, $N$ is at least 54.

To construct an example where $N$ equals 54, it is sufficient to fill the table with numbers row by row: first, the first row with numbers from 1 to 18, the second row with numbers from 19 to 36, and the third row with numbers from 37 to 54, and the rest can be placed arbitrarily. Note that the average numbers in the rows will be the numbers from 37 to 54 (each of them is greater than any of the numbers from 1 to 36 and less than any of the numbers from 55 to 90). The number 54 is the largest of them.