Ukrajina 2008

Let's consider rectangles with their sides along the grid lines. On the board let there be a rectangle with dimensions $A\times B$, which has no painted sectors ($A$ is its width, $B$ is its height). According to the problem statement $A$ columns which the rectangle occupies must have $A$ painted sectors. On the other hand, these $A$ sectors can not be in those $B$ rows that the rectangle occupies. The rest is $(10-B)$ rows. The obligatory condition is $10-B\ge A$ (otherwise we are not able to place $A$ painted sectors in $(10-B)$ rows, one at a row). $A+B\le 10$ (you'll obtain the same result if you consider $B$ rows). Thus rectangle $5\times 5$ has the maximal area (this area $A(10-A)\le (\frac{A+10-A}{2}{)}^2=25$ is represented in the Cauchy inequality or as a parabola with its branches down). We can easily draw an example of the board with such a rectangle (fig.3).

Fig.3