RU-ALLRUSSIAN 2015 Problem 118

Given natural numbers $a$ and $b$ , such that $a<b<2a$ . Some cells on a graph are colored such that in every rectangle with dimensions $A \times B$ or $B \times A$ , at least one cell is colored. For which greatest $\alpha$ can you say that for every natural number $N$ you can find a square $N \times N$ in which at least $\alpha \cdot N^2$ cells are colored?