5. In an $n \times n$ table of cells, there are "十" and "-". It is allowed to take a cell and change the signs in the column and row passing through this cell (at the same time, the sign of the cell itself also changes). Find all $n$ for which, from any initial arrangement of signs in the table, one can transition to a table of all plus signs.

## Senior League
5. Answer: for even $n$.

Let $n$ be odd. We will color the $n \times n$ board in a checkerboard pattern so that the center is black. It is easy to see that on each cross (union of a row and a column) there will be an even number of white cells. Then the parity of the number of signs on white cells will be an invariant.

On the other hand, for even $n$, take a cross $K_{C}$ with the center in cell $C$. $C_{i} \neq C (i=1, \ldots, 2 n-2)$ are the other cells of this cross. Then if we perform the operation of replacing signs in $K_{C}$, as well as in all $K_{C_{i}}$ simultaneously, the result will be that in each cell except $C$ the signs will change an even number of times, and in cell $C$ - an odd number. Thus, we have learned how to change the sign in only one cell.

## Senior League