## Task A-2.4.

In each field of a $10 \times 10$ table, a natural number is written, and all 20 sums of the numbers in its rows and columns are mutually different. What is the smallest possible sum of all the numbers in such a filled table?
## Solution.

There are 10 rows and 10 columns, making a total of 20 different sums. The smallest possible sums of rows and columns are $10,11,12, \ldots, 29$.

Since adding all the sums of the numbers both by rows and by columns gives us twice the sum of all the numbers in the table, the theoretically smallest sum of all the numbers in such a table is

$$
\frac{1}{2}\left(\frac{29 \cdot 30}{2}-\frac{9 \cdot 10}{2}\right)=195
$$

1 point

To ensure that these sums are actually achievable, we must construct at least one example of the arrangement of numbers. One possible example is given in the following table:

| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 4 | 5 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 6 | 7 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 8 | 9 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 | 9 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 11 |

Note: Writing the beginning of the table construction (e.g., two rows or columns) is worth 2 points. Constructing any table that meets the conditions of the problem is worth 4 points. Different tables, other than permutations of the rows and columns of the existing table, can be obtained by arranging different $2 \times 2$ blocks on the diagonal so that the sums of the numbers in the rows and columns form two consecutive numbers (the other numbers are still set to 1). One such example is the table below.

| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 5 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 6 | 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 9 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 10 | 3 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 14 | 3 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 17 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 18 | 3 |