Found problems: 68
1989 ITAMO, 3
Prove that, for every tetrahedron $ABCD$, there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.
1967 IMO, 2
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
2000 ITAMO, 3
A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.
1978 Czech and Slovak Olympiad III A, 4
Is there a tetrahedron $ABCD$ such that $AB+BC+CD+DA=12\text{ cm}$ with volume $\mathrm V\ge2\sqrt3\text{ cm}^3?$
1966 IMO Shortlist, 21
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?
1987 Polish MO Finals, 4
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.
1984 IMO Shortlist, 13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2019 Korea USCM, 2
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 $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.
1976 Czech and Slovak Olympiad III A, 6
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 $ABCD'.$
1994 Czech And Slovak Olympiad IIIA, 2
A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?
1993 ITAMO, 6
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 $C'$.
1970 IMO Shortlist, 5
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 $PQRS$).
1967 IMO Longlists, 40
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
1993 Austrian-Polish Competition, 2
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.
1979 Spain Mathematical Olympiad, 7
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.
2010 All-Russian Olympiad Regional Round, 11.6
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 $OSDA$ .
2002 Paraguay Mathematical Olympiad, 2
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]
1983 Putnam, B1
Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.
1999 Czech And Slovak Olympiad IIIA, 2
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.
Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.
1967 IMO Shortlist, 3
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 $\alpha, \beta, \gamma.$
1996 Spain Mathematical Olympiad, 6
A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.
1996 German National Olympiad, 3
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 $V$
1986 Polish MO Finals, 2
Find the maximum possible volume of a tetrahedron which has three faces with area $1$.
2014 BMT Spring, 13
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.
1953 Polish MO Finals, 3
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.