Through each vertex of a tetrahedron with a given volume $ V $, a plane is drawn parallel to the opposite face of the tetrahedron. Calculate the volume of the tetrahedron formed by these planes.

Matrices $A$, $B$ are given as follows. \[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\] Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.

Find the maximum possible volume of a tetrahedron which has three faces with area $1$.

Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.

In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$.

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .

Given two regular tetrahedrons with edges of length $\sqrt2$, transforming into one another with central symmetry. Let $\Phi$ be the set the midpoints of segments whose ends belong to different tetrahedrons. Find the volume of the figure $\Phi$.

A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.

A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.

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.