A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$. KöMaL A.723. (April 2018), G. Kós

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).

A super ball rolling on the floor enters a half circular track (radius $R$). The ball rolls without slipping around the track and leaves (velocity $v$) traveling horizontally in the opposite direction. Afterwards, it bounces on the floor. How far (horizontally) from the end of the track will the ball bounce for the second time? The ball’s surface has a theoretically infinite coefficient of static friction. It is a perfect sphere of uniform density. All collisions with the ground are perfectly elastic and theoretically instantaneous. Variations could involve the initial velocity being given before the ball enters the track or state that the normal force between the ball and the track right before leaving is zero (centripetal acceleration). [i]Problem proposed by Brian Yue[/i]

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.)

Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.

A right circular cone of volume $ A$, a right circular cylinder of volume $ M$, and a sphere of volume $ C$ all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then $ \textbf{(A)}\ A \minus{} M \plus{} C \equal{} 0 \qquad \textbf{(B)}\ A \plus{} M \equal{} C \qquad \textbf{(C)}\ 2A \equal{} M \plus{} C$ $ \textbf{(D)}\ A^2 \minus{} M^2 \plus{} C^2 \equal{} 0 \qquad \textbf{(E)}\ 2A \plus{} 2M \equal{} 3C$

a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere. b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) $ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $

Let $A,B,C,D,E,F,G$ and $H$ be eight pairwise distinct points on the surface of a sphere. The quadruples $(A,B,C,D), (A,B,F,E),(B,C,G,F),(C,D,H,G)$ and $(D,A,E,H)$ of points are coplanar. Prove that the quadruple $(E,F,G,H)$ is coplanar aswell.

What is the greatest number of parts that $5$ spheres can divide the space into?

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

Define a $\emph{big circle}$ in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose $(AB)$ as the big circle that passes through $A$ and $B$. Also, let a $\emph{Spheric Triangle}$ be $3$ connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also $\angle XYZ$ the angle between $(XY)$ and $(YZ)$. Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S. Let $\Delta ABC$ be a spheric triangle with all its angles $<90^{\circ}$ such that there is a circle $\omega$ tangent to $(BC)$,$(CA)$,$(AB)$ in $D,E,F$. Show that there is $P\in S$ with $\angle PAB=\angle DAC$, $\angle PCA=\angle FCB$, $\angle PBA=\angle EBC$.

There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.

A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle? (It is not obligatory to use all segments. The side of the triangle can be formed by two segments)

A right rectangular prism is inscribed within a sphere. The total area of all the faces [of] the prism is $88$, and the total length of all its edges is $48$. What is the surface area of the sphere? $\text{(A) }40\pi\qquad\text{(B) }32\pi\sqrt{2}\qquad\text{(C) }48\pi\qquad\text{(D) }32\pi\sqrt{3}\qquad\text{(E) }56\pi$

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

Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas. [list](i) Prove that all the discs that he cuts have integral areas. (ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]

Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. [i]L. Lovasz[/i]

On a table are a cone, resting on the base, and six equal spheres tangent to the cone. Besides that, each sphere is tangent to the two adjacent spheres. Knowing that the radius $R$ of the base of the cone is half its height and determine the radius $r$ of the spheres.

The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$

This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.