Found problems: 2265
Kyiv City MO 1984-93 - geometry, 1989.10.5
The base of the quadrangular pyramid $SABCD$ is a quadrilateral $ABCD$, the diagonals of which are perpendicular. The apex of the pyramid is projected at intersection point $O$ of the diagonals of the base. Prove that the feet of the perpendiculars drawn from point $O$ to the side faces of the pyramid lie on one circle.
2008 AMC 12/AHSME, 18
Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$?
$ \textbf{(A)}\ \sqrt{85} \qquad
\textbf{(B)}\ \sqrt{90} \qquad
\textbf{(C)}\ \sqrt{95} \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ \sqrt{105}$
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 $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
2009 Polish MO Finals, 5
A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.
1969 Bulgaria National Olympiad, Problem 6
It is given that $r=\left(3\left(\sqrt6-1\right)-4\left(\sqrt3+1\right)+5\sqrt2\right)R$ where $r$ and $R$ are the radii of the inscribed and circumscribed spheres in a regular $n$-angled pyramid. If it is known that the centers of the spheres given coincide,
(a) find $n$;
(b) if $n=3$ and the lengths of all edges are equal to a find the volumes of the parts from the pyramid after drawing a plane $\mu$, which intersects two of the edges passing through point $A$ respectively in the points $E$ and $F$ in such a way that $|AE|=p$ and $|AF|=q$ $(p<a,q<a)$, intersects the extension of the third edge behind opposite of the vertex $A$ wall in the point $G$ in such a way that $|AG|=t$ $(t>a)$.
1981 All Soviet Union Mathematical Olympiad, 326
The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.
1987 IMO Shortlist, 9
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?
[i]Proposed by Hungary.[/i]
[hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]
2000 Austrian-Polish Competition, 2
In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.
1952 Moscow Mathematical Olympiad, 217
Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.
2008 AMC 12/AHSME, 18
A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid?
$ \textbf{(A)}\ 392 \qquad
\textbf{(B)}\ 196\sqrt{6} \qquad
\textbf{(C)}\ 392\sqrt2 \qquad
\textbf{(D)}\ 392\sqrt3 \qquad
\textbf{(E)}\ 784$
2009 AMC 12/AHSME, 20
A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have?
$ \textbf{(A)}\ 200\qquad
\textbf{(B)}\ 2n\qquad
\textbf{(C)}\ 300\qquad
\textbf{(D)}\ 400\qquad
\textbf{(E)}\ 4n$
2021 Adygea Teachers' Geometry Olympiad, 4
Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.
2015 BMT Spring, Tie 3
The permutohedron of order $3$ is the hexagon determined by points $(1, 2, 3)$, $(1, 3, 2)$, $(2, 1, 3)$, $(2, 3, 1)$, $(3, 1, 2)$, and $(3, 2, 1)$. The pyramid determined by these six points and the origin has a unique inscribed sphere of maximal volume. Determine its radius.
2004 Estonia Team Selection Test, 6
Call a convex polyhedron a [i]footballoid [/i] if it has the following properties.
(1) Any face is either a regular pentagon or a regular hexagon.
(2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it).
Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.
1969 Polish MO Finals, 5
For which values of n does there exist a polyhedron having $n$ edges?
2000 Harvard-MIT Mathematics Tournament, 1
The sum of $3$ real numbers is known to be zero. If the sum of their cubes is $\pi^e$, what is their product equal to?
2009 Estonia Team Selection Test, 3
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
2014 NIMO Summer Contest, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
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 Jozsef Wildt International Math Competition, W. 59
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).
2007 Iran MO (3rd Round), 7
A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles.
[img]http://i18.tinypic.com/6cdmvi8.png[/img]
a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated.
[img]http://i19.tinypic.com/4qgx30j.png[/img]
b) What's the answer if 1 is replaced with 0?
1985 Bulgaria National Olympiad, Problem 3
A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.
2017 International Zhautykov Olympiad, 3
Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$
2004 AMC 12/AHSME, 20
Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac{5}{16} \qquad
\textbf{(C)}\ \frac38 \qquad
\textbf{(D)}\ \frac{7}{16} \qquad
\textbf{(E)}\ \frac12$
1967 IMO Shortlist, 6
Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).