1. Each cell of a $5 \times 6$ table is colored in one of three colors: blue, red, or yellow. In each row of the table, the number of red cells is not less than the number of blue cells and not less than the number of yellow cells, and in each column of the table, the number of blue cells is not less than the number of red cells and not less than the number of yellow cells. How many yellow cells can there be in such a table? Provide an example of a corresponding coloring.
Answer: 6.

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

Solution. Since in each row there are no fewer red cells than blue ones, there are no fewer of them in the entire table. Then the number of red and blue cells is the same in each column. Indeed, if in one of the columns there are fewer red cells than blue ones, then there will be fewer of them in the entire table, since in the other columns there are no more blue ones. In addition, the number of blue cells in each column is no less than $\frac{5}{3}$, that is, it is at least 2. Therefore, the table contains at least 12 blue cells and as many red ones. Therefore, there are no more than 6 yellow cells in the table. On the other hand, in each column the total number of red and blue cells is even, so there is at least one yellow cell in it. Therefore, the table contains no fewer than 6 yellow cells. An example of coloring with six yellow cells is shown in the figure.