Olympiad problems

1983 Spain Mathematical Olympiad, 7

A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

1967 IMO, 2

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

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

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.

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 $PQRS$).

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.

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: 3d geometry , sphere , volume , geometry

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 $\alpha, \beta, \gamma.$

1987 Polish MO Finals, 4

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

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.

2003 Portugal MO, 1

Tags: geometry , 3d geometry , cube , volume

The planet Caramelo is a cube with a $1$ km edge. This planet is going to be wrapped with foam anti-gluttons in order to prevent the presence of greedy ships less than $500$ meters from the planet. What the minimum volume of foam that must surround the planet?

1994 Czech And Slovak Olympiad IIIA, 2

Tags: cube , polyhedron , volume , max , 3d geometry , projection , geometry

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 $M$?

1999 Czech And Slovak Olympiad IIIA, 2

Tags: 3d geometry , tetrahedron , volume , geometry

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.

1935 Moscow Mathematical Olympiad, 006

Tags: geometry , 3d geometry , pyramid , volume

The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

Champions Tournament Seniors - geometry, 2011.4

Tags: 3d geometry , volume , pyramid , geometry

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).

1981 Brazil National Olympiad, 6

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

The centers of the faces of a cube form a regular octahedron of volume $V$. Through each vertex of the cube we may take the plane perpendicular to the long diagonal from the vertex. These planes also form a regular octahedron. Show that its volume is $27V$.

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 $\alpha, \beta, \gamma.$

1993 Austrian-Polish Competition, 2

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

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.

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.

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 $\frac{2}{3 \pi}.$