Bogdanov I.I.

In some cells of a $10 \times 10$ table, several crosses and several zeros are placed. It is known that there is no line (row or column) completely filled with the same symbols (crosses or zeros). However, if any symbol is placed in any empty cell, this condition will be violated. What is the minimum number of symbols that can be in the table?
Let us assume that we have filled the table according to the condition, and cell $A$ is free. Since placing any symbol in it should result in a line of identical symbols, there must be a line containing it where all other cells are filled with crosses, and the same is true for noughts (these lines must, of course, be a row and a column). In particular, all free cells are in different rows and columns. Suppose there are more than two empty cells. Then for two of them (say, for $A$ and $B$), the directions of the line of crosses coincide (that is, either for both the crosses are in their horizontals, or for both - in their verticals) (see fig.). Let the crosses be in their columns, and the noughts - in their rows; then at the intersection of the row containing $A$ with the column containing $B$, there must be both a cross and a nought (since this intersection is not empty); a contradiction. Therefore, there cannot be more than two empty cells. An example with two empty cells is shown in the figure (instead of asterisks, any symbols can be placed).

| $A$ | 0 | 0 | $?$ | $\circ$ | 0 | 0 | 0 | 0 | 0 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $?$ | 0 | 0 | $B$ | $\circ$ | 0 | 0 | 0 | 0 | 0 |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |
| $\times$ |  |  | $\times$ |  |  |  |  |  |  |


|  | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |
| $\times$ | $\circ$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ |

## Answer

98.

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