10. Natural numbers are inscribed in the cells of a $100 \times 100$ square such that all 200 sums in the rows (rows and columns) are distinct. What is the smallest possible sum of all the numbers in the table?
- Answer: 19950. For convenience, we will assume that the numbers can be zeros (we will add one to each number later). Then the sum in each row is at least 0, and the minimum possible values of these sums range from 0 to

| 0 |  |  |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 2 |  |  |  |  |  |
|  | 0 | 4 |  |  |  |  |
|  |  | 1 | 6 |  |  |  |
|  |  |  |  | $\ddots$ |  |  |
|  |  |  |  |  | 196 |  |
|  |  |  |  |  | 1 | 198 |

Table 1: to problem 10.

199, and the minimum possible sum is $199 \cdot 200 / 4 = 9950$. We will show that there exists an example where exactly these sums are achieved. For this, we will place even numbers from zero to 198 on the main diagonal, and ones in every other cell on the secondary diagonal (Table 1). All other numbers will be declared zeros. Now, to get an example for the original problem, we will add one to each number in the table.

## (D. Maksimov)