ILL 1982 Problem 57

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 $4.$