IMO 1992 LL CAN7

Origin: CAN

Problem

Let X be a bounded, nonempty set of points in the Cartesian plane. Let f(X) be the set of all points that are at a distance of at most 1 from some point in X. Let f n(X) = f(f(. . . (f(X)) . . . )) (n times). Show that f n(X) becomes “more circular” as n gets larger. In other words, if rn = sup{radii of circles contained in f n(X)} and Rn = inf{radii of circles containing f n(X)}, then show that Rn/rn gets arbitrarily close to 1 as n becomes arbitrarily large.