There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?

Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

Consider a solid hemisphere of radius $1$. Find the distance from its center of mass to the base.

A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.

A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 4\sqrt5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 6\sqrt3$

A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal. (M. Volchkevich)

Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others. [i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]

Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.

Given a sphere of radius $ R $ and a plane $ \alpha $ having no common points with this sphere. A point $ S $ moves in the plane $ \alpha $, which is the vertex of a cone tangent to the sphere along a circle with center $ C $. Find the locus of point $ C $. [hide=another is Polish MO 1967 p6] [url=https://artofproblemsolving.com/community/c6h3388032p31769739]here[/url][/hide]

Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.

Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.

Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.

The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?

Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.

Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area? (Problem from Latvia)

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of $3$ from the indicated vertex and at distances $\sqrt5, \sqrt6, \sqrt7$ from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$