Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.

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 $ABCD'.$

Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$. Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$.

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.

In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png[/img]

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 $PQRS$).

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 $OSDA$ .

Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior $x^2 +y^2 +z^2 +t^2 <r^2$

Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$

A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments as shown below. This cup has height $h$ and volume $V$. A second cup that is exactly the same shape as the first is held upright inside the first cup so that its base is a distance of $\tfrac{h}2$ from the base of the first cup. The volume of liquid that will t inside the first cup and outside the second cup can be written $\tfrac{m}{n}\cdot V$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] pair s = (10,1); draw(ellipse((0,0),4,1)^^ellipse((0,-6),2,.5)); fill((3,-6)--(-3,-6)--(0,-2.1)--cycle,white); draw((4,0)--(2,-6)^^(-4,0)--(-2,-6)); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-2.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); pair s = (10,-2); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-4.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); //darn :([/asy]

Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra. [asy] import graph; size(12.57cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */ pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1); draw((6,3.5)--(8,1.5), zzzzff); draw((7,3)--(5,1), blue); draw((6,3.5)--(7,3), blue); draw((6,3.5)--(5,1), blue); draw((5,1)--(8,1.5), blue); draw((7,3)--(8,1.5), blue); draw((4,3.5)--(2,1.5), ffzzzz); draw((1,3)--(2,1.5), ffqqtt); draw((2,1.5)--(3,1), ffqqtt); draw((1,3)--(3,1), ffqqtt); draw((4,3.5)--(1,3), ffqqtt); draw((4,3.5)--(3,1), ffqqtt); draw((-3,3)--(-3,1), linewidth(1.6)); draw((-3,3)--(-1,3), linewidth(1.6)); draw((-1,3)--(-1,1), linewidth(1.6)); draw((-3,1)--(-1,1), linewidth(1.6)); draw((-3,3)--(-2,3.5), linewidth(1.6)); draw((-2,3.5)--(0,3.5), linewidth(1.6)); draw((0,3.5)--(-1,3), linewidth(1.6)); draw((0,3.5)--(0,1.5), linewidth(1.6)); draw((0,1.5)--(-1,1), linewidth(1.6)); draw((-3,1)--(-2,1.5)); draw((-2,1.5)--(0,1.5)); draw((-2,3.5)--(-2,1.5)); draw((1,3)--(1,1), linewidth(1.6)); draw((1,3)--(3,3), linewidth(1.6)); draw((3,3)--(3,1), linewidth(1.6)); draw((1,1)--(3,1), linewidth(1.6)); draw((1,3)--(2,3.5), linewidth(1.6)); draw((2,3.5)--(4,3.5), linewidth(1.6)); draw((4,3.5)--(3,3), linewidth(1.6)); draw((4,3.5)--(4,1.5), linewidth(1.6)); draw((4,1.5)--(3,1), linewidth(1.6)); draw((1,1)--(2,1.5)); draw((2,3.5)--(2,1.5)); draw((2,1.5)--(4,1.5)); draw((5,3)--(5,1), linewidth(1.6)); draw((5,3)--(6,3.5), linewidth(1.6)); draw((5,3)--(7,3), linewidth(1.6)); draw((7,3)--(7,1), linewidth(1.6)); draw((5,1)--(7,1), linewidth(1.6)); draw((6,3.5)--(8,3.5), linewidth(1.6)); draw((7,3)--(8,3.5), linewidth(1.6)); draw((7,1)--(8,1.5)); draw((5,1)--(6,1.5)); draw((6,3.5)--(6,1.5)); draw((6,1.5)--(8,1.5)); draw((8,3.5)--(8,1.5), linewidth(1.6)); label("$ A $",(-3.4,3.41),SE*labelscalefactor); label("$ D $",(-2.16,4.05),SE*labelscalefactor); label("$ H $",(-2.39,1.9),SE*labelscalefactor); label("$ E $",(-3.4,1.13),SE*labelscalefactor); label("$ F $",(-1.08,0.93),SE*labelscalefactor); label("$ G $",(0.12,1.76),SE*labelscalefactor); label("$ B $",(-0.88,3.05),SE*labelscalefactor); label("$ C $",(0.17,3.85),SE*labelscalefactor); label("$ A $",(0.73,3.5),SE*labelscalefactor); label("$ B $",(3.07,3.08),SE*labelscalefactor); label("$ C $",(4.12,3.93),SE*labelscalefactor); label("$ D $",(1.69,4.07),SE*labelscalefactor); label("$ E $",(0.60,1.15),SE*labelscalefactor); label("$ F $",(2.96,0.95),SE*labelscalefactor); label("$ G $",(4.12,1.67),SE*labelscalefactor); label("$ H $",(1.55,1.82),SE*labelscalefactor); label("$ A $",(4.71,3.47),SE*labelscalefactor); label("$ B $",(7.14,3.10),SE*labelscalefactor); label("$ C $",(8.14,3.82),SE*labelscalefactor); label("$ D $",(5.78,4.08),SE*labelscalefactor); label("$ E $",(4.6,1.13),SE*labelscalefactor); label("$ F $",(6.93,0.96),SE*labelscalefactor); label("$ G $",(8.07,1.64),SE*labelscalefactor); label("$ H $",(5.65,1.90),SE*labelscalefactor); dot((-3,3),dotstyle); dot((-3,1),dotstyle); dot((-1,3),dotstyle); dot((-1,1),dotstyle); dot((-2,3.5),dotstyle); dot((0,3.5),dotstyle); dot((0,1.5),dotstyle); dot((-2,1.5),dotstyle); dot((1,3),dotstyle); dot((1,1),dotstyle); dot((3,3),dotstyle); dot((3,1),dotstyle); dot((2,3.5),dotstyle); dot((4,3.5),dotstyle); dot((4,1.5),dotstyle); dot((2,1.5),dotstyle); dot((5,3),dotstyle); dot((5,1),dotstyle); dot((6,3.5),dotstyle); dot((7,3),dotstyle); dot((7,1),dotstyle); dot((8,3.5),dotstyle); dot((8,1.5),dotstyle); dot((6,1.5),dotstyle); [/asy]

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.

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.

Let $S$ be the set of all tetrahedra which satisfy: (1) the base has area $1$, (2) the total face area is $4$, and (3) the angles between the base and the other three faces are all equal. Find the element of $S$ which has the largest volume.

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.

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

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?

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]

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 cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.

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?

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.

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 $C'$.

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 $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.

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.