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Hellenic Mathematical Olympiad ARCHIMEDES

Geometry Difficulty 4.7 AIME Prove it Greece

Is it possible to put on the plane $2014$ points, so that with vertices from these points to construct $1006^2$ parallelograms of area $1$ ?

Solution

We shall prove that it is possible. We consider two parallel lines $\varepsilon_1, \varepsilon_2$ 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 $\varepsilon_1$ $1006$ segments of length $1$ and in the line $\varepsilon_2$ 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 $1$ .

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