9. Several rooks have beaten all the white cells of a $40 \times 40$ chessboard. What is the maximum number of black cells that could remain unbeaten? (A rook beats the cell it stands on.)
$\square$ Answer: 400.

Evaluation. If there is an unpicked cell, then on the rows (vertical and horizontal) passing through it, there are no rooks. To attack all cells of one color on such a row, rooks must be placed on at least 20 perpendicular rows. Thus, there are no more than 20 rows in each direction that are free from rooks. Unattacked cells are located at the intersections of such rows, so there are no more than $20^{2}$ of them.

Example. We can assume that the bottom-left cell is black. Place 20 rooks on white cells along the left and bottom edges. Then, at the intersections of the unattacked rows, all cells are black, totaling 400, and all white cells are attacked.

(A. Shapovalov)

## Fourth Round