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fn(X) becomes more circular - IMO LongList 1992 CAN4

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September 2, 2010
functiongeometrycoordinate geometryIterationIMO ShortlistIMO Longlist

Problem Statement

Let XX be a bounded, nonempty set of points in the Cartesian plane. Let f(X)f(X) be the set of all points that are at a distance of at most 11 from some point in XX. Let fn(X)=f(f(⋯(f(X))⋯ ))f_n(X) = f(f(\cdots(f(X))\cdots)) (nn times). Show that fn(X)f_n(X) becomes “more circular” as nn gets larger. In other words, if rn=sup⁡{radii of circles contained in fn(X)}r_n = \sup\{\text{radii of circles contained in } f_n(X) \} and Rn=inf⁡{radii of circles containing fn(X)}R_n = \inf \{\text{radii of circles containing } f_n(X)\}, then show that Rn/rnR_n/r_n gets arbitrarily close to 11 as nn becomes arbitrarily large.
I'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.
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