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 $n^{2}$.

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic. [i] Tibor Bakos and Géza Kós [/i]

A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(150); real t=0.12; void ball(pair x, real r, real h, bool ww=true) { pair xx=yscale(t)*x+(0,h); path P=circle(xx,r); unfill(P); draw(P); if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r)); } pair X=(0,0); real H=17, h=5, R=h/2; draw(H*dir(120)--(0,0)--H*dir(60)); draw(ellipse((0,0.87*H),H/2,t*H/2)); pair Y=(R,h+2*R),C=(0,h); real r; for(int k=0;k<20;++k) { r=-(dir(30)*Y).x; Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3; } ball(Y.x*dir(90),r,Y.y,false); ball(X,R,h); ball(Y.x*dir(-30),r,Y.y); ball(Y.x*dir(210),r,Y.y);[/asy]

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.

The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.

Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$. $ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$. $ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.

We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane. Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

Let $n$ be a positive integer. An $n-\emph{simplex}$ in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1,\cdots , P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $\mathcal{S}$ we denote by $v(\mathcal{S})$ the volume of $\mathcal{S}$, and we write $C(\mathcal{S})$ for the center of the unique sphere containing all the vertices of $\mathcal{S}$. Suppose that $P$ is a point inside an $n$-simplex $\mathcal{S}$. Let $\mathcal{S}_i$ be the $n$-simplex obtained from $\mathcal{S}$ by replacing its $i^{\text{th}}$ vertex by $P$. Prove that : \[ \sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S}) \]

[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous. [b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.

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 $n>c(P)/V^2.$

Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

A tetrahedron has five edges of length $3$ and circumradius $2$. What is the length of the sixth edge?

Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?

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.

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$. Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

$A,B$, 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 $S$.