IMO 1969 LL BUL9

One hundred convex polygons are placed on a square with edge

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IMO 1969 LL BUL9

Origin: BUL

Problem

One hundred convex polygons are placed on a square with edge of length 38 cm. The area of each of the polygons is smaller than \pi cm2, and the perimeter of each of the polygons is smaller than 2\pi cm. Prove that there exists a disk with radius 1 in the square that does not intersect any of the polygons.