1. Each cell of a $5 \times 5$ table is colored in one of three colors: blue, red, or yellow. In each row of the table, the number of yellow cells is not less than the number of red cells and not less than the number of blue cells, and in each column of the table, the number of red cells is not less than the number of yellow cells and not less than the number of blue cells. How many blue cells can there be in such a table? Provide an example of a corresponding coloring.
Answer: 5.

| $\mathrm{C}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{K}$ | $\mathrm{C}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ |
| $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{C}$ | $\mathrm{K}$ | $\mathrm{K}$ |
| $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{C}$ | $\mathrm{K}$ |
| $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{K}$ | $\mathrm{C}$ |

Solution. Since in each row the number of yellow cells is not less than the number of red cells, the same is true for the entire table. Therefore, the number of yellow and red cells is the same in each column. Indeed, if in one of the columns the number of yellow cells is less than the number of red cells, then there will be fewer of them in the entire table, since in the other columns their number is not greater than the number of red cells. Moreover, the number of red cells in each column is not less than $\frac{5}{3}$, that is, it is at least 2. Thus, the table contains at least 10 red cells and the same number of yellow cells. Therefore, there are no more than 5 blue cells in the table. On the other hand, in each column the total number of yellow and red cells is even, so there is at least one blue cell in each column. Thus, the table contains no fewer than 5 blue cells. An example of coloring with five blue cells is shown in the figure.