2. For what largest natural $\mathrm{k}$ can the cells of a $5 \times 5$ table be filled with zeros and ones (each cell must contain exactly one number) so that there are $k$ rows, in each of which the sum of the numbers is at least 3, and $k$ columns, in each of which the sum of the numbers is at most 2? (O. Nechaeva, I. Rubanov)
Answer. For $k=4$. Solution. For $k=5$, it is impossible to fill the table in the required manner, since when calculating the sum of the numbers in the table by adding the sums of the rows, it turns out that it is not less than $3 \cdot 5=15$, while when calculating the same sum by the columns, it turns out that it is not

| 1 | 1 | 0 | 0 | 1 |
| :--- | :--- | :--- | :--- | :--- |
| 1 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 |

more than $2 \cdot 5=10$. An example of filling the table that satisfies the condition of the problem for $k=4$ is shown on the right.