Found problems: 68
1965 Czech and Slovak Olympiad III A, 4
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) $\frac13 < x \leq\frac12$.
1990 IMO Longlists, 27
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.
1990 IMO Shortlist, 10
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.
1977 Czech and Slovak Olympiad III A, 6
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.
2013 Purple Comet Problems, 28
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]
1985 Austrian-Polish Competition, 6
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 $ABCD$.
1970 IMO Longlists, 20
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 $PQRS$).
1966 IMO Longlists, 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?
2012 Purple Comet Problems, 26
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]
1998 All-Russian Olympiad Regional Round, 11.7
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 $\Phi$.
1937 Moscow Mathematical Olympiad, 034
Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments
1995 Romania Team Selection Test, 2
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
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).
2007 Sharygin Geometry Olympiad, 20
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.
2004 Harvard-MIT Mathematics Tournament, 9
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?
1935 Moscow Mathematical Olympiad, 006
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.
2018 Polish Junior MO First Round, 7
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.
1971 IMO Longlists, 49
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.
1967 IMO Longlists, 32
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 $\alpha, \beta, \gamma.$
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.
1997 Estonia National Olympiad, 3
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.
VI Soros Olympiad 1999 - 2000 (Russia), 11.8
Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.
II Soros Olympiad 1995 - 96 (Russia), 11.7
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.
1989 Poland - Second Round, 3
Given is a trihedral angle $ OABC $ with a vertex $ O $ and a point $ P $ in its interior. Let $ V $ be the volume of a parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Calculate the minimum volume of a tetrahedron whose three faces are contained in the faces of the trihedral angle $OABC$ and the fourth face contains the point $P$.
2015 Finnish National High School Mathematics Comp, 2
The lateral edges of a right square pyramid are of length $a$. Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$. Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.