11.8. Given natural numbers $a$ and $b$, such that $a < b < 2a$. On a grid plane, some cells are marked in such a way that in every grid rectangle $a \times b$ or $b \times a$ there is at least one marked cell. For what largest $\alpha$ can we assert that for any natural number $N$ there exists a grid square $N \times N$ in which at least $\alpha N^{2}$ cells are marked?

(I. Bogdanov, O. Podlipsky)
Answer. $\alpha=1 /\left(2 a^{2}-2 a b+b^{2}\right)$.

Solution. Introduce a coordinate system on the plane so that the centers of the cells, and only they, have integer coordinates. We will say that a cell has the same coordinates as its center. We will call a rectangle $a \times b$ vertical or horizontal if its side of length $b$ is vertical or horizontal, respectively.