Olympiad problems

1970 IMO Shortlist, 5

Tags: geometry , 3d geometry , tetrahedron , volume , equation

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 $PQRS$).

1997 Estonia National Olympiad, 3

Tags: 3d geometry , tetrahedron , volume , geometry , sphere , inscribed , circumscribed

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.

1996 German National Olympiad, 3

Tags: 3d geometry , tetrahedron , volume , geometry

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

Tags: volume , tetrahedron , 3d geometry , geometry

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

2005 Abels Math Contest (Norwegian MO), 1b

Tags: geometry , 3d geometry , pyramid , even , volume , integer

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.

1984 IMO Shortlist, 13

Tags: 3d geometry , tetrahedron , volume , geometric inequality , geometry

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

1984 IMO Longlists, 11

Tags: geometry , 3d geometry , tetrahedron , volume , geometric inequality

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

1993 ITAMO, 6

Tags: rotation , geometry , 3d geometry , volume , cube

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 $C'$.

2007 Sharygin Geometry Olympiad, 20

Tags: geometry , 3d geometry , pyramid , volume , maximum

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

1971 IMO, 2

Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume

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.

1978 Czech and Slovak Olympiad III A, 4

Tags: geometry , 3d geometry , tetrahedron , volume

Is there a tetrahedron $ABCD$ such that $AB+BC+CD+DA=12\text{ cm}$ with volume $\mathrm V\ge2\sqrt3\text{ cm}^3?$

1949 Putnam, A2

Tags: vector , volume

We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.

1967 IMO Longlists, 32

Tags: geometry , 3d geometry , sphere , volume

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 $\alpha, \beta, \gamma.$

1971 IMO Shortlist, 16

Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume

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.

1995 Romania Team Selection Test, 2

Tags: parallelepiped , combinatorial geometry , combinatorics , volume , cube , partition

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.

Champions Tournament Seniors - geometry, 2011.4

Tags: 3d geometry , geometry , volume , pyramid

The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).

1971 IMO Longlists, 49

Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume

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.

2018 Polish Junior MO First Round, 7

Tags: 3d geometry , square , volume , plane , geometry

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

1996 Spain Mathematical Olympiad, 6

Tags: geometry , regular polygon , volume , 3-dimensional geometry

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.

II Soros Olympiad 1995 - 96 (Russia), 11.7

Tags: geometry , 3d geometry , prism , volume

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

1960 Poland - Second Round, 6

Tags: geometry , 3d geometry , volume , tetrahedron

Calculate the volume of the tetrahedron $ ABCD $ given the edges $ AB = b $, $ AC = c $, $ AD = d $ and the angles $ \measuredangle CAD = \beta $, $ \measuredangle DAB = \gamma $ and $ \measuredangle BAC = \delta$.

1967 IMO Shortlist, 3

Tags: geometry , 3d geometry , sphere , volume

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 $\alpha, \beta, \gamma.$

1990 IMO Longlists, 27

Tags: geometry , 3d geometry , ellipse , sphere , volume

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.

1971 IMO Longlists, 50

Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume

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.

1966 IMO Longlists, 21

Tags: geometry , 3d geometry , area , volume , geometric inequality

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?