IMO 1987 LL POL49

Origin: POL

Problem

In the coordinate system in the plane we consider a convex polygon W and lines given by equations x = k, y = m, where k and m are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of W intersects a square if the boundary contains an interior point of the square. Prove that the boundary of W intersects at most 4\lceild\rceilunit squares, where d is the maximal distance of points belonging to W (i.e., the diameter of W) and \lceild\rceilis the least integer not less than d.