Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.

A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that \[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

What is the radius of the largest sphere that fits inside an octahedron of side length $1$?

Given is a convex polyhedron with $ k $ faces $ S_1, \ldots, S_k $. Let us denote the vector of length 1 perpendicular to the wall $ S_i $ ($ i = 1, \ldots, k $) directed outside the given polyhedron by $ \overrightarrow{n_i} $, and the surface area of this wall by $ P_i $. Prove that $$ \sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.$$

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$

A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.

a) A lamp of a lighthouse enlights an angle of $90$ degrees. Prove that you can turn the lamps of four arbitrary posed lighthouses so, that all the plane will be enlightened. b) There are eight lamps in eight points of the space. Each can enlighten an octant (three-faced space polygon with three mutually orthogonal edges). Prove that you can turn them so, that all the space will be enlightened.

In regular $n$-regular pyramid, the range value of dihedral angle of two adjacent sides is $\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)$

Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.

$PABCDE$ is a pyramid with $ABCDE$ a convex pentagon. A plane meets the edges $PA, PB, PC, PD, PE$ in points $A', B', C', D', E'$ distinct from $A, B, C, D, E$ and $P$. For each of the quadrilaterals $ABB'A', BCC'B, CDD'C', DEE'D', EAA'E'$ take the intersection of the diagonals. Show that the five intersections are coplanar.

A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

[asy] import olympiad; import cse5; size(5cm); pointpen = black; pair A = Drawing((10,17.32)); pair B = Drawing((0,0)); pair C = Drawing((20,0)); draw(A--B--C--cycle); pair X = 0.85*A + 0.15*B; pair Y = 0.82*A + 0.18*C; pair W = (-11,0) + X; pair Z = (19, 9); draw(W--X, EndArrow); draw(X--Y, EndArrow); draw(Y--Z, EndArrow); anglepen=black; anglefontpen=black; MarkAngle("\theta", C,Y,Z, 3); [/asy] The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number. [i](Ahaan Rungta, 5 points)[/i]

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.