The polyhedron $PABCDQ$ has the form shown in the figure. It is known that $ABCD$ is parallelogram, the planes of the triangles of the $PAC$ and $PBD$ mutually perpendicular, and also mutually perpendicular are the planes of triangles $QAC$ and $QBC$. Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces. [img]https://1.bp.blogspot.com/-UM5PKEGGWqc/X1V2cXAFmwI/AAAAAAAAMdw/V-Qr94tZmqkj3_q-5mkSICGF1tMu-b_VwCLcBGAsYHQ/s0/2007.5%2Bchampions%2Btourn.png[/img]

[asy]size(100); draw((0,0)--(1,0)--(1,1)--(1,2)--(2,2)--(2,3)--(2,4)--(1,4)--(1,3)--(0,3)--(-1,3)--(-1,2)--(0,2)--(0,1)--cycle); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,2)--(0,3)); draw((1,2)--(1,3)); draw((1,3)--(2,3)); label("Z",(0.5,0.2),N); label("X",(0.5,1.2),N); label("V",(0.5,2.2),N); label("U",(-0.5,2.2),N); label("W",(1.5,2.2),N); label("Y",(1.5,3.2),N);[/asy] A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $ \text{X}$ is: \[ \textbf{(A)}\ \text{Z} \qquad \textbf{(B)}\ \text{U} \qquad \textbf{(C)}\ \text{V} \qquad \textbf{(D)}\ \text{W} \qquad \textbf{(E)}\ \text{Y} \]

Space is dissected into congruent cubes. Is it necessarily true that for each cube there exists another cube so that both cubes have a whole face in common?

Consider a piece of material in the shape of a right circular conical frustum with radii $R,r,R>r$. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii $R',r'$ of the cavity. Decide for which ratio $R/r$ the problem has a solution. [img]https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png[/img]

[b]a)[/b] Let's consider a finite number of big circles of a sphere that do not pass all from a point. Show that there exists such a point that is found only in two of the circles. (With big circle we understand the circles with radius equal to the radius of the sphere.) [b]b)[/b] Using the result of part $a)$ show that, for a set of $n$ points in a plane, that are not all in a line, there exists a line that passes through only two points of the given set.

In the tetrahedron $ABCD$, perpendiculars $AB'$, $AC'$, $AD'$ are dropped from vertex $A$, on the plane dividing the dihedral angles at the edges $CD$, $BD$, $BC$ in half. Prove that the plane $(B'C'D' )$ is parallel to the plane $(BCD)$.

Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

[b]rolling cube[/b] $a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. time allowed for this question was 1 hour.

Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.

In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? $ \textbf{(A) }105 \qquad \textbf{(B) }112\frac{1}{2} \qquad \textbf{(C) }120 \qquad \textbf{(D) }135 \qquad \textbf{(E) }150 \qquad $

A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is $\textbf {(A) } 54 \qquad \textbf {(B) } 72 \qquad \textbf {(C) } 76 \qquad \textbf {(D) } 84\qquad \textbf {(E) } 86$

Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$, where $a,b,c$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d$.

There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? $ \textbf{(A)}\ 54 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 96 $

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

Let $ABCDA_1B_1C_1D_1$ be a cube. On the sides $AB{}$ and $AD{}$ there are the points $M{}$ and $N{}$, respectively, such that $AM+AN=AB$. Show that the measure of the dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$ doe not depend on the positions of $M{}$ and $N{}$. Find this measure.

The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD = 10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.

The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$. Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.