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Four circles are drawn.  Let $A_1,$ $A_2,$ $A_3,$ $A_4$ be the areas of the regions, so  $A_1$ is the area inside the smallest circle, $A_2$ is the area outside the smallest circle and inside the second-smallest circle, and so on.  The areas satisfy
\[A_1 = \frac{A_2}{2} = \frac{A_3}{6}\]
Let $r_1$ denote the radius of the smallest circle, and let $r_4$ denote the radius of the largest circle.  Find $\frac{r_4}{r_1}.$

 Dec 17, 2023

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