In a right circular cone with the radius of the base $R$ and the height $h$ are $n$ spheres of the same radius $r$ ($n \ge 3$). Each ball touches the base of the cone, its side surface and other two balls. Determine $r$.

Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

The sphere inscribed in tetrahedron $ABCD$ touches the sides $ABD$ and $DBC$ at points $K$ and $M$, respectively. Prove that $\angle AKB = \angle DMC$.

Six line segments of lengths $17, 18, 19, 20, 21$ and $23$ form the side edges of a triangular pyramid (also called a tetrahedron). Can there exist a sphere tangent to all six lines?

For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.) [i]Z. Szabo[/i]

Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?

We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.

Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.

The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid. [asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

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]

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? $ \textbf{(A)}\ 2: 1 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 4: 1 \qquad \textbf{(D)}\ 16: 3 \qquad \textbf{(E)}\ 6: 1$

Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

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)

Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

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 real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that \[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\] Prove that the points $E,F,G,H,K,L$ all lie on a sphere.

Each point of the sphere of radius $r\ge 1$ is colored in one of $n$ colors ($n \ge 2$), and for each color there is a point on the sphere colored in this color. Prove that there are points $A_i$, $B_i$, $i= 1, ..., n$ on the sphere such that the colors of the points $A_1, ..., A_n$ are pairwise different and the color of the point $B_i$ at a distance of $1$ from $A_i$ is different from the color of the point $A_1, i= 1, ..., n$