Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that $$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$

Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

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.

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.

The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism? [asy] // djmathman diagram unitsize(1cm); defaultpen(linewidth(0.7)+fontsize(11)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype ("4 4")); dot("$A$",A,N); dot("$B$",B,1.2*N); dot("$C$",C,N); dot("$D$",D,dir(0)); dot("$E$",E,S); dot("$F$",F,1.5*S); dot("$G$",G,S); dot("$H$",H,W); dot("$I$",I,NE); dot("$J$",J,1.5*S); [/asy] $\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288\qquad$

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height. [asy] string s = "1010101010111111"; defaultpen(linewidth(0.7)); for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); } }} label("South View",(2,4),N); s = "0101110111111111"; for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { x=x+5; draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); x=x-5; } }} label("East View",(7,4),N);[/asy]

Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide: a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere, b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur.com/08FbQs.png[/img] size(140); defaultpen(linewidth(.7)); real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3); path hex=rotate(alpha)*polygon(6); pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha)); pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y); int i; pair[] H; for(i=0; i<6; i=i+1) { H[i] = dir(alpha+60*i);} fill(X--Y--Z--cycle, rgb(204,255,255)); fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255)); fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153)); fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255)); draw(hex^^X--Y--Z--cycle); draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5")); draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

Let $ABCD$ be a three-link broken line in space, all links of which are equal and $\angle BCD=90^o$. Find the distance from $A$ to the midpoint of $BD$, if $AD = a$.

A wooden cube with edge length $ n$ units (where $ n$ is an integer $ >2$) is painted black all over. By slices parallel to its faces, the cube is cut into $ n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $ n$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ \text{none of these}$

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.

a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines? b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?

Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?

The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.

The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron. (S. Shosman, O. Ogievetsky)

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.

(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)? (b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?

Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("$1$",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("$2$",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("$3$",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("$4$",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("$5$",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("$6$",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("$7$",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("$8$",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("$9$",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$