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Geometric inequality on radii of concentric circles

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September 28, 2010
inequalitiescirclesgeometrygeometric inequalityIMO ShortlistIMO Longlist

Problem Statement

Two concentric circles have radii RR and rr respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between 32⋅R+rR−r−1\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1 and 6320⋅R+rR−r.\frac{63}{20} \cdot \frac{R+r}{R-r}.
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