[ Numerical tables and their properties ]

In table $A$ of size $10 \times 10$, some numbers are written. Let the sum of all numbers in the first row be denoted by $s_{1}$, in the second row by $s_{2}$, and so on. Similarly, the sum of numbers in the first column is denoted by $t_{1}$, in the second column by $t_{2}$, and so on.

A new table $B$ of size $10 \times 10$ is constructed, with numbers written as follows: in the first cell of the first row, the smallest of the numbers $s_{1}$ and $t_{1}$ is written, in the third cell of the fifth row, the smallest of the numbers $s_{5}$ and $t_{3}$ is written, and so on for the entire table. It turns out that the cells of table $B$ can be numbered from 1 to 100 in such a way that the number in the cell with the $k$-th number is less than or equal to $k$. What is the maximum value that the sum of all numbers in table $A$ can take under these conditions?
Example of table $A$, for which the sum of all numbers is 955.

| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 91 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 92 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 93 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 94 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 95 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 97 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 98 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 99 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 |

In this case, table $B$ coincides with $A$.

Estimate. By renumbering the columns and rows and, possibly, reflecting the table relative to the diagonal, we can ensure that $t_{10}$ is the largest of the numbers $s_{1}, \ldots, s_{10}, t_{1}, \ldots, t_{10}$. Then in the tenth column stand the numbers $s_{1}, \ldots, s_{10}$. The sum of the numbers in these cells does not exceed the sum of the cell numbers in which they stand, and therefore is no more than $91+\ldots+100 = 955$. On the other hand, this sum is equal to the sum of all the numbers in table $A$. Therefore, the sum of all the numbers in table $A$ does not exceed 955.

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