Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
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Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
Suppose two of these circles with centers $C_1$ and $C_2$ touch each other. for the angle $\alpha:=C_1OC_2$, we have $n=[\frac{2\pi}\alpha ]$ and $sin\frac\alpha 2=\frac{R-r}{R+r}$. So \[n\le\frac{\pi}{\frac\alpha 2}\le\frac{\pi}{sin\frac\alpha 2}=\pi \frac{R+r}{R-r}\le\frac{63}{20} \cdot \frac{R+r}{R-r}.\]
For the left ineq, it suffices to show $\arcsin\frac{R-r}{R+r}\le2\frac{\sqrt R -\sqrt r }{\sqrt R +\sqrt r }$ or $\arcsin\frac{1-x^2}{1+x^2}\le2\frac{1-x }{1+x }$ for $0<x<1$ which is easy by calculus.