3-5. In space, there are 3 planes and a sphere. In how many different ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, it is actually about the tangency of spheres, i.e., it is not assumed that the spheres can only touch externally - ed. note.)[^0]
Solve 5. Answer: from 0 to 16 (depending on the arrangement of the planes and the sphere).

Let $O$ and $R$ be the center and radius of the given sphere $S$. Suppose that the sphere $S_{1}$ with center $O_{1}$ touches the given sphere and three given planes. We associate with the sphere $S_{1}$ the sphere $S_{1}^{\prime}$ with center $O_{1}$ and radius $O_{1} O$. The sphere $S_{1}^{\prime}$ passes through the given point $O$ and touches three planes, each of which is at a distance $R$ from the given planes. For each plane, there are exactly two planes at a distance $R$. Therefore, the sphere $S_{1}^{\prime}$ passes through the given point $O$ and touches a triplet of planes, and there are $2^{3}=8$ different such triplets.

It is easy to verify that there are no more than two spheres passing through a given point and touching three given planes. In the case where the three given planes intersect at one point, this is proven as follows. Inscribing a sphere in the trihedral angle formed by the given planes and containing the given point, we draw a line connecting the given point and the point of intersection of the given planes. The desired spheres correspond to the points of intersection of this line with the given sphere, and there are no more than two such points. The case where the given point lies on one of the given planes is considered separately. Additionally, two simpler cases need to be considered: 1) there are two parallel planes and a third plane intersects them; 2) the lines of intersection of the planes are parallel. (It is easy to see that if all three planes are parallel or intersect along one line, then a sphere cannot touch all three such planes.)

In the end, we obtain that there are no more than 16 spheres $S_{1}^{\prime}$. The sphere $S_{1}$ can be reconstructed from the sphere $S_{1}^{\prime}$ non-uniquely. To reconstruct it, we need to take the point $O^{\prime}$ where the line $O O_{1}$ intersects the sphere $S$; the sphere $S_{1}$ is the sphere with center $O_{1}$ and radius $O_{1} O^{\prime}$. There are two points of intersection of the sphere with the line passing through its center, so we get two spheres. But only one of them can touch the three given planes (the second sphere touches other planes). Therefore, there are no more than 16 spheres that touch the given sphere and three given planes.

Now we will show that for different arrangements of the given planes and the given sphere, the number of spheres that touch them can be any integer from 0 to 16. We will assume that the given planes intersect along three parallel lines, and the distances between these lines are pairwise different. Spheres that touch the three given planes sweep out four cylinders corresponding to the inscribed and three exscribed circles of the triangle formed by the intersection of the given planes with a plane orthogonal to all three given planes. These cylinders do not have common points, including points of tangency.

Consider a cylinder and a point outside it. Consider the family of spheres $S_{R}$ with center at the chosen point. We are interested in spheres inscribed in the cylinder and touching the sphere $S_{R}$. If the radius $R$ of the sphere $S_{R}$ is small, then there are no such spheres. We will increase the radius $R$ until the sphere $S_{R}$ touches the cylinder. In this case, the number of desired spheres is 1. We will again increase the radius $R$. When the sphere $S_{R}$ intersects the cylinder, the number of desired spheres becomes 2. Then the sphere $S_{R}$ will intersect the cylinder and also touch it at one point. In this case, the number of desired spheres is 3. If the radius $R$ is increased even more, the number of desired spheres will be 4.

These observations show that if we choose a point outside the four given cylinders and consider the family of spheres $S_{R}$ with center at the chosen point, then by gradually increasing the radius $R$, we will obtain configurations for which the number of desired spheres changes sequentially from 0 to 16. We just need to ensure that two points of tangency of the sphere $S_{R}$ with the cylinders do not appear simultaneously.

# III Olympiad (1937)

## First Round