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

[b](a)[/b] Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum. [b](b)[/b] Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.

A uniform pool ball of radius $r$ and mass $m$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $-1 \le \beta \le 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass $m$ and radius $r$ is $I_{cm} = \frac{2}{5}mr^2$.) [asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.8)); draw((-3,-1)--(3,-1)); draw((-2.4,0.1)--(-2.4,0.6),EndArrow); draw((-2.5,0)--(2.5,0),dashed); draw((-2.75,0.7)--(-0.8,0.7),EndArrow); label("$J$",(-2.8,0.7),W); label("$\beta r$",(-2.3,0.35),E); draw((0,-1.5)--(0,1.5),dashed); draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow); label("$r$",(1.75,-0.5),E); [/asy] (a) Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$. (b) For what value of $\beta$ does the ball immediately begin to roll without slipping, regardless of the value of $\mu$?

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .

Given a sphere, a great circle of the sphere is a circle on the sphere whose diameter is also a diameter of the sphere. For a given positive integer $n,$ the surface of a sphere is divided into several regions by $n$ great circles, and each region is colored black or white. We say that a coloring is good if any two adjacent regions (that share an arc as boundary, not just a finite number of points) have different colors. Find, with proof, all positive integers $n$ such that in every good coloring with $n$ great circles, the sum of the areas of the black regions is equal to the sum of the areas of the white regions.

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

Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively. Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$. Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.

The surface areas of the bases of a given truncated triangular pyramid are equal to $ B_1 $ and $ B_2 $. This pyramid can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. Prove that the lateral surface area of the given pyramid is $ (\sqrt{B_1} + \sqrt{B_2})(\sqrt[4]{B_1} + \sqrt[4]{B_2})^2 $.

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

Three spheres are constructed so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.

Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls.

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.

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

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 $.

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

$S_{1},S_{2},S_{3}$ 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.

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.