10. (ICE 3) Let $S_{1}$ and $S_{2}$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_{1}$ and $S_{2}$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
10. Denote by \( r, R \) (take w.l.o.g. \( r < R \)) the radii and by \( A, B \) the centers of the spheres \( S_{1}, S_{2} \) respectively. Let \( s \) be the common radius of the spheres in the ring, \( C \) the center of one of them, say \( S \), and \( D \) the foot of the perpendicular from \( C \) to \( AB \). The centers of the spheres in the ring form a regular \( n \)-gon with center \( D \), and thus \( \sin (\pi / n) = s / CD \). Using Heron's formula on the triangle \( ABC \), we obtain \( (r+R)^{2} CD^{2} = 4 r R s (r+R+s) \), and hence 
![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-502.jpg?height=565&width=438&top_left_y=1118&top_left_x=865)
$$ \sin ^{2} \frac{\pi}{n} = \frac{s^{2}}{CD^{2}} = \frac{(r+R)^{2} s}{4(r+R+s) r R} $$
Choosing the unit of length so that \( r+R=2 \), for simplicity of writing, we write (1) as \( 1 / \sin ^{2}(\pi / n) = r R (1 + 2 / s) \). Let now \( v \) be half the angle at the top of the cone. Then clearly \( R - r = (R + r) \sin v = 2 \sin v \), giving us \( R = 1 + \sin v, r = 1 - \sin v \). It follows that
$$ \frac{1}{\sin ^{2} \frac{\pi}{n}} = \left(1 + \frac{2}{s}\right) \cos ^{2} v $$
We need to express \( s \) as a function of \( R \) and \( r \). Let \( E_{1}, E_{2}, E \) be collinear points of tangency of \( S_{1}, S_{2} \), and \( S \) with the cone. Obviously, \( E_{1} E_{2} = E_{1} E + E_{2} E \), i.e., \( 2 \sqrt{r s} + 2 \sqrt{R s} = 2 \sqrt{R r} = (R + r) \cos v = 2 \cos v \). Hence,
$$ \cos ^{2} v = s (\sqrt{R} + \sqrt{r})^{2} = s (R + r + 2 \sqrt{R r}) = s (2 + 2 \cos v) $$
Substituting this into (2), we obtain \( 2 + \cos v = 1 / \sin (\pi / n) \). Therefore \( 1 / 3 < \sin (\pi / n) < 1 / 2 \), and we conclude that the possible values for \( n \) are 7, 8, and 9.