10,11

Three spheres of radius $R$ touch each other and a certain plane. Find the radius of the sphere that touches the given spheres and the same plane.
Let $A, B$ and $C$ be the points of tangency of the spheres of radius $R$ with the given plane, and $D$ be the point of tangency with this plane of the sphere of the desired radius $r$ (Fig.1). Consider the orthogonal projection of the spheres onto the given plane (Fig.2). We obtain three mutually tangent circles with centers $A, B$ and $C$ of radii $R$ and a circle with center $D$ of radius $r$, and $D$ is the center of the equilateral triangle $A B C$. Therefore,

$$
A D=\frac{A B}{\sqrt{3}}=\frac{2 R}{\sqrt{3}}
$$

On the other hand, since $A D$ is the common tangent of the tangent spheres of radii $R$ and $r$, then $A D=2 \sqrt{R r}$. From the equation $2 \sqrt{R r}=\frac{2 R}{\sqrt{3}}$ we find that $r=\frac{R}{3}$.

## Answer

$\frac{R}{3}$