Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

Find the equation of the smallest sphere which is tangent to both of the lines $$\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} t+1\\ 2t+4\\ -3t +5 \end{pmatrix},\;\;\;\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} 4t-12\\ -t+8\\ t+17 \end{pmatrix}.$$

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.

A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky

In space, a point $O$ and a finite set of vectors $ \overrightarrow{v_1},\ldots,\overrightarrow{v_n} $ are given . We consider the set of points $ P $ for which the vector $ \overrightarrow{OP} $can be represented as a sum $ a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n} $with coefficients satisfying the inequalities $ 0 \leq a_i \leq 1 $ $( i = 1, 2, \ldots, n $). Decide whether this set can be a tetrahedron.

Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point? [i]A. Chukhnov[/i]

It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane ($14$ values in total) take two different values. The smallest is $1$. What can the edge of a cube be equal to?

Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$

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 $\angle PAB=\angle DAC$, $\angle PCA=\angle FCB$, $\angle PBA=\angle EBC$.

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

The sum of $3$ real numbers is known to be zero. If the sum of their cubes is $\pi^e$, what is their product equal to?

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric. [i]Proposed by: Géza Kós, Budapest[/i]

Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.

The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.

Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.

We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.

Let $C$ be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners at $C,$ keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces indentical. If at the conclusion of this process the fourteen faces so have the same area, find the area of each of face of the new solid.

$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$. Find the locus of $X$. What happens to $X$ as $M$ tends to (1) $D$, (2) $C$? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

Given a tetrahedron such that product of the opposite edges is $ 1$. Let the angle between the opposite edges be $ \alpha$, $ \beta$, $ \gamma$, and circumradii of four faces be $ R_1$, $ R_2$, $ R_3$, $ R_4$. Prove that \[ \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}} \]