Found problems: 2265
2024 German National Olympiad, 2
Six quadratic mirrors are put together to form a cube $ABCDEFGH$ with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex $A$ in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube.
For each positive integer $n$, determine the set of vertices where the laser beam can leave the cube after exactly $n$ reflections.
2012 Putnam, 2
Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$
2019 Polish Junior MO First Round, 7
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/721989193ffd830fd7ad43bdde7e177c942c76.png[/img]
2002 AIME Problems, 2
Three vertices of a cube are $P=(7,12,10),$ $Q=(8,8,1),$ and $R=(11,3,9).$ What is the surface area of the cube?
Kyiv City MO 1984-93 - geometry, 1985.10.2
Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?
2013 Baltic Way, 13
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.
2005 District Olympiad, 4
Prove that no matter how we number the vertices of a cube with integers from 1 to 8, there exists two opposite vertices in the cube (e.g. they are the endpoints of a large diagonal of the cube), united through a broken line formed with 3 edges of the cube, such that the sum of the 4 numbers written in the vertices of this broken lines is at least 21.
2011 NIMO Problems, 5
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
[i]Proposed by Lewis Chen[/i]
1992 Czech And Slovak Olympiad IIIA, 2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
2018 CCA Math Bonanza, T8
A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there?
[i]2018 CCA Math Bonanza Team Round #8[/i]
1972 IMO, 3
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
1991 Arnold's Trivium, 77
Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.
2018 Moscow Mathematical Olympiad, 2
There is tetrahedron and square pyramid, both with all edges equal $1$. Show how to cut them into several parts and glue together from these parts a cube (without voids and cracks, all parts must be used)
2005 International Zhautykov Olympiad, 3
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.
2014 USAMTS Problems, 4:
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.
1983 IMO Shortlist, 13
Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?
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.
1999 Croatia National Olympiad, Problem 2
The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.
2010 Moldova Team Selection Test, 4
Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n\minus{}2)}4\plus{}2$ cubes.
2013 Princeton University Math Competition, 8
Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular. For $R$ the maximum possible radius of this sphere, find $R^2$.
1991 Vietnam National Olympiad, 3
Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.
2003 Iran MO (3rd Round), 18
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.
1962 Vietnam National Olympiad, 4
Let be given a tetrahedron $ ABCD$ such that triangle $ BCD$ equilateral and $ AB \equal{} AC \equal{} AD$. The height is $ h$ and the angle between two planes $ ABC$ and $ BCD$ is $ \alpha$. The point $ X$ is taken on $ AB$ such that the plane $ XCD$ is perpendicular to $ AB$. Find the volume of the tetrahedron $ XBCD$.
2008 Harvard-MIT Mathematics Tournament, 1
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
1994 Moldova Team Selection Test, 9
Let $O{}$ be the center of the circumscribed sphere of the tetrahedron $ABCD$. Let $L,M,N$ respectively be the midpoints of the segments $BC,CA,AB$. It is known that $AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$. Find their value.