Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.

In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.

We say that a rational number is [i]spheric[/i] if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that: [b]a)[/b] $ 7 $ is not spheric. [b]b)[/b] a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.

Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.

Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively. Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.

The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 35\qquad \textbf{(E)}\ 99$

A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then (1) all tangent points of these faces to the sphere would coincide with one point, $H$, and (2) the vertices of the faces would lie on a circle centered at $H$.

Given a tetrahedron $ABCD.$ The sphere $\omega$ inscribed in it touches the face $ABC$ at point $T$. Sphere $\omega' $ touches face $ABC$ at point $T'$ and extensions of faces $ABD$, $BCD$, $CAD$. Prove that the lines $AT$ and $AT'$ are symmetric wrt bisector of angle $\angle BAC$

A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. [i]Original formulation:[/i] A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

Given a triangular pyramid $SABC$, in which $\angle BSC = \alpha$, $\angle CSA =\beta$, $\angle ASB = \gamma$, and the dihedral angles at the edges $SA$ and $SB$ have the value of $\phi$ and $\delta$, respectively. Prove that $\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.$$

Let $S$ be a given sphere with center $O$ and radius $r$. Let $P$ be any point outside then sphere $S$, and let $S'$ be the sphere with center $P$ and radius $PO$. Denote by $F$ the area of the surface of the part of $S'$ that lies inside $S$. Prove that $F$ is independent of the particular point $P$ chosen.

Some identically sized spheres are piled in $n$ layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of $n^2$ spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer $m$ such that as $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sqrt{m}}{\pi}$. Find $m$. [center][img]https://snag.gy/bIwyl6.jpg[/img][/center]

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

In regular triangular pyramid $S-ABC$, hieght $SO=3$, length of sides of bottom surface is $6$. Projection of $A$ on plane $SBC$ is $O'$. $P\in AO',\frac{AP}{PO'}=8$. Draw a plane parallel to plane $ABC$ and passes $P$. Find the area of the cross section.

A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. [i]Original formulation:[/i] A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$+$",(0,0)); draw(shift(1,0)*box); label("$-$",(1,0)); draw(shift(2,0)*box); label("$+$",(2,0)); draw(shift(3,0)*box); label("$-$",(3,0)); draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$+$",(1,0.8)); draw(shift(2,0.8)*box); label("$+$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2)); [/asy] $\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$

Color the vertexes of a quadrangular pyramid in one color, satisfying that two end points of any edge are in different colors. We have only 5 colors, then the number of ways coloring the quadrangular pyramid is________.

An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color. [i]M. Smurov[/i]

Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point $(1,1,1)$ and (b) if the tangent plane is drawn at any point $P$ and $X,Y, Z$ are the intersections of this plane with the $x, y$ and $z-$axis respectively, then $P$ is the orthocenter of the triangle $XYZ.$