What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.

A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.

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

Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.

* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.

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 $AA'$).

Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.

(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one. (b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.

What is the greatest number of parts into which the plane can be cut by the edges of $ n $ squares?

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

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.

Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 26$

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$

In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.

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 $A$.

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.

The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?

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 $

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

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?