Two spheres of one radius and two of another are arranged so that each sphere touches three others and one plane. Find the ratio of the radius of the larger sphere to the radius of the smaller one.

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Let $O_1$ and $O_2$ be the centers of the larger spheres with radius $R$, and $A$ and $B$ be their points of contact with a given plane; $O_3$ and $O_4$ be the centers of the smaller spheres with radius $r$, and $C$ and $D$ be their points of contact with the given plane. Then $ACBD$ is a rhombus (Fig.1) with side length $2 \sqrt{r R}$ (Fig.2) and diagonals $2r$ and $2R$. Therefore, $AC = \sqrt{r^2 + R^2}$. From the equation

$$
2 \sqrt{r R} = \sqrt{r^2 + R^2}
$$

we find that $\frac{R}{r} = 2 \pm \sqrt{3}$. Since $\frac{R}{r} > 1$, the desired ratio is $2 + \sqrt{3}$.

## Answer

$2 + \sqrt{3}$

## Problem