Found problems: 27
1962 Polish MO Finals, 6
Given three lines $ a $, $ b $, $ c $ pairwise skew. Is it possible to construct a parallelepiped whose edges lie on the lines $ a $, $ b $, $ c $?
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]
1998 Tournament Of Towns, 5
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)
1996 Romania National Olympiad, 3
Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.
2019 District Olympiad, 3
Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.
1980 All Soviet Union Mathematical Olympiad, 299
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
1991 Poland - Second Round, 6
The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.
1973 Polish MO Finals, 3
A polyhedron $W$ has the following properties:
(i) It possesses a center of symmetry.
(ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram.
(iii) There is a vertex of $W$ at which exactly three edges meet.
Prove that $W$ is a parallelepiped.
1995 Belarus Team Selection Test, 2
There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.
2011 German National Olympiad, 2
The price for sending a packet (a rectangular cuboid) is directly proportional to the sum of its length, width, and height. Is it possible to reduce the cost of sending a packet by putting it into a cheaper packet?
2009 Bulgaria National Olympiad, 3
Through the points with integer coordinates in the right-angled coordinate system $Oxyz$ are constructed planes, parallel to the coordinate planes and in this way the space is divided to unit cubes. Find all triples ($a, b, c$) consisting of natural numbers ($a \le b \le c$) for which the cubes formed can be coloured in $abc$ colors in such a way that every palellepiped with dimensions $a \times b \times c$, having vertices with integer coordinates and sides parallel to the coordinate axis doesn't contain unit cubes in the same color.
1973 All Soviet Union Mathematical Olympiad, 188
Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?
1996 Poland - Second Round, 6
Prove that every interior point of a parallelepiped with edges $a,b,c$ is on the distance at most $\frac12 \sqrt{a^2 +b^2 +c^2}$ from some vertex of the parallelepiped.
2009 Tournament Of Towns, 4
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.
1966 Polish MO Finals, 3
Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.
1999 Romania National Olympiad, 3
Let $ABCDA'B'C'D'$ be a right parallelepiped, $E$ and $F$ the projections of $A$ on the lines $A'D$, $A'C$, respectively, and $P, Q$ the projections of $B'$ on the lines $A'C'$ and $A'C$ Prove that
a) the planes $(AEF)$ and $(B'PQ)$ are parallel
b) the triangles $AEF$ and $B'PQ$ are similar.
1995 All-Russian Olympiad Regional Round, 11.2
A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.
1985 Tournament Of Towns, (096) 5
A square is divided into rectangles.
A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$.
(a) Prove that every two rectangles in such a division are members of a certain "chain".
(b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") .
(A.I . Golberg, V.A. Gurevich)
2019 District Olympiad, 2
Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$.
a) Prove that $BM + BN + BP = BD'$.
b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.
Kyiv City MO Seniors 2003+ geometry, 2004.11.4
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be the feet of the perpendiculars drawn from point $A$ on the lines $A_1D$ and $A_1C$, respectively, and the points $P$ and $Q$ be the feet of the perpendiculars drawn from point $B_1$ on the lines $A_1C_1$ and $A_1C$, respectively. Prove that $\angle EFA = \angle PQB_1$
2021 Yasinsky Geometry Olympiad, 3
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$.
a) Draw a section of a parallelepiped with plane $\gamma$.
b) Justify what geometric figure is this section, and find its area.
(Alexander Shkolny)
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.
1985 All Soviet Union Mathematical Olympiad, 411
The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?
III Soros Olympiad 1996 - 97 (Russia), 11.5
All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses. Plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$. It is known that $A_1K = a$, $A_1M = b$, and$ a + b$ is an edge of the parallelepiped. Prove that the plane $AKM$ touches the sphere inscribed in the parallelepiped. Let us denote by $Q$ the touchpoint of this sphere with the plane $AKM $. In what ratio does the straight line $AQ$ divide the segment $KM$?
1997 Romania National Olympiad, 3
$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$.
a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value.
b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$.
c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.