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Geometric inequality on radii of concentric circles

Source:

9/28/2010
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}.
inequalitiescirclesgeometrygeometric inequalityIMO ShortlistIMO Longlist