Hellenic Mathematical Olympiad ARCHIMEDES

We shall prove that it is possible. We consider two parallel lines ${\epsilon }_1,{\epsilon }_2$ with distance $1$. We put on each of them $1007$ points such that every two successive of them have distance $1$. Then we have in the line ${\epsilon }_1$ $1006$ segments of length $1$ and in the line ${\epsilon }_2$ we have also $1006$ unit segments. Any pair of unit segments having members from both lines gives rise to a parallelogram of area $1$. Therefore we have totally $1006^2$ parallelograms of area $1$.