IMO Shortlist 1982 problem 2

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Dodao/la: arhiva
2. travnja 2012.
Let K be a convex polygon in the plane and suppose that K is positioned in the coordinate system in such a way that
\text{area } (K \cap Q_i) =\frac 14  \text{area } K \  (i = 1, 2, 3, 4, ),
where the Q_i denote the quadrants of the plane. Prove that if K contains no nonzero lattice point, then the area of K is less than 4.
Izvor: Međunarodna matematička olimpijada, shortlist 1982