A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? $ \textbf{(A) }105 \qquad \textbf{(B) }112\frac{1}{2} \qquad \textbf{(C) }120 \qquad \textbf{(D) }135 \qquad \textbf{(E) }150 \qquad $

Four spheres, each of radius $r$, lie inside a regular tetrahedron with side length $1$ such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find $r$.

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.

A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points, $A$ and $B$, such that $AB=42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere.

Given a sphere $K$, determine the set of all points $A$ that are vertices of some parallelograms $ABCD$ that satisfy $AC \le BD$ and whose entire diagonal $BD$ is contained in $K$.

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$. [asy] import three; import solids; size(8cm); //currentprojection = perspective(1, 1, 10); triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3); draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white); draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white); draw(surface(sphere(1,20)), white); label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20)); dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8); dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2); dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]

$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.

Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$. (a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$. (b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

Consider two spheres $\Sigma_1$ and $\Sigma_2$ and a line $\Delta$ not meeting them. Let $C_i$ and $r_i$ be the center and radius of $\Sigma_i$, and let $H_i$ and $d_i$ be the orthogonal projection of $C_i$ onto $\Delta$ and the distance of $C_i$ from $\Delta~(i=1,2)$. For a point $M$ on $\Delta$, let $\delta_i(M)$ be the length of a tangent $MT_i$ to $\Sigma_i$, where $T_i\in\Sigma_i~(i=1,2)$. Find $M$ on $\Delta$ for which $\delta_1(M)+\delta_2(M)$ is minimal.

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.

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.

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.)

In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.

Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.

Given a tetrahedron $ ABCD $ whose edges $ AB, BC, CD, DA $ are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.