Found problems: 2265
2017 AMC 10, 11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2003 Croatia National Olympiad, Problem 3
In a tetrahedron $ABCD$, all angles at vertex $D$ are equal to $\alpha$ and all dihedral angles between faces having $D$ as a vertex are equal to $\phi$. Prove that there exists a unique $\alpha$ for which $\phi=2\alpha$.
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 $\Phi$.
2000 Bundeswettbewerb Mathematik, 3
For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.
2002 Putnam, 2
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
1999 Croatia National Olympiad, Problem 1
For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.
II Soros Olympiad 1995 - 96 (Russia), 11.5
$6$ points are taken on the surface of the sphere, forming three pairs of diametrically opposite points on the sphere. Consider a convex polyhedron with vertices at these points. Prove that if this polyhedron has one right dihedral angle, then it has exactly $6$ right dihedral angles.
1967 IMO Longlists, 23
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
\[af^2 + bfg +cg^2 \geq 0\]
holds if and only if the following conditions are fulfilled:
\[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]
II Soros Olympiad 1995 - 96 (Russia), 10.5
Is there a six-link broken line in space that passes through all the vertices of a given cube?
2013 All-Russian Olympiad, 2
The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.
2023 Belarusian National Olympiad, 9.8
On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different.
Find the smallest possible amount of the sum of all written numbers.
Champions Tournament Seniors - geometry, 2013.3
On the base of the $ABC$ of the triangular pyramid $SABC$ mark the point $M$ and through it were drawn lines parallel to the edges $SA, SB$ and $SC$, which intersect the side faces at the points $A1_, B_1$ and $C_1$, respectively. Prove that $\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le \sqrt{SA+SB+SC}$
2015 USAMTS Problems, 2
A net for a polyhedron is cut along an edge to give two [b]pieces[/b]. For example, we may cut a cube net along the red edge to form two pieces as shown.
[asy]
size(5.5cm);
draw((1,0)--(1,4)--(2,4)--(2,0)--cycle);
draw((1,1)--(2,1));
draw((1,2)--(2,2));
draw((1,3)--(2,3));
draw((0,1)--(3,1)--(3,2)--(0,2)--cycle);
draw((2,1)--(2,2),red+linewidth(1.5));
draw((3.5,2)--(5,2));
filldraw((4.25,2.2)--(5,2)--(4.25,1.8)--cycle,black);
draw((6,1.5)--(10,1.5)--(10,2.5)--(6,2.5)--cycle);
draw((7,1.5)--(7,2.5));
draw((8,1.5)--(8,2.5));
draw((9,1.5)--(9,2.5));
draw((7,2.5)--(7,3.5)--(8,3.5)--(8,2.5)--cycle);
draw((11,1.5)--(11,2.5)--(12,2.5)--(12,1.5)--cycle);
[/asy]
Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? If you think the answer is no, prove that no pair of polyhedra can result in the same two pairs of pieces. If you think the answer is yes, provide an example; a clear example will suffice as a proof.
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
1986 IMO Longlists, 30
Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.
2007 Princeton University Math Competition, 2
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?
IV Soros Olympiad 1997 - 98 (Russia), 11.6
On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)
2008 Mexico National Olympiad, 2
We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices.
a) Prove that $\frac23E\le V$.
b) Can $E=V$?
1984 IMO Longlists, 65
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
2013 Uzbekistan National Olympiad, 5
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
1979 Polish MO Finals, 2
Prove that the four lines, joining the vertices of a tetrahedron with the incenters of the opposite faces, have a common point if and only if the three products of the lengths of opposite sides are equal.
2011 Sharygin Geometry Olympiad, 25
Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?
2011 Purple Comet Problems, 15
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
2011 AMC 12/AHSME, 9
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{4}{9} \qquad
\textbf{(D)}\ \frac{5}{9} \qquad
\textbf{(E)}\ \frac{2}{3} $
May Olympiad L1 - geometry, 2010.1
A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.