The number of regular triangles that three apexes are among eight vertex of a cube is $\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

Prove that if in a tetrahedron $ ABCD $ the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point, then $$AB \cdot CD = AC \cdot BD = AD \cdot BC$$ and that the converse also holds.

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?

A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.) [i](8 points)[/i] [i]Mikhail Evdokimov[/i]

A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$. [i](Proposed by Ahaan Rungta)[/i]

How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?

Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to (a) no less than $ 5$ faces? (b) exactly $5$ faces? (G. Galperin)

Is there a six-link broken line in space that passes through all the vertices of a given cube?

If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is: $\text{(A)} \ 2 \qquad \text{(B)} \ -2-\frac{3i\sqrt{3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ -\frac{3i\sqrt{3}}{4} \qquad \text{(E)} \ -2$

The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?

The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.

In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable?

Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$, $VB$, $VC$ and $VD$ at $M$, $N$, $P$, $Q$ respectively. Find $VQ : QD$, if $VM : MA = 2 : 1$, $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$.