Found problems: 2265
1948 Moscow Mathematical Olympiad, 152
a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections.
b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.
2009 Romanian Master of Mathematics, 3
Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that
\[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3,
\]
denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel.
[i]Nikolai Ivanov Beluhov, Bulgaria[/i]
2014 Uzbekistan National Olympiad, 5
Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $.
Find the volume of the pyramid.
1973 IMO Shortlist, 9
Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?
IV Soros Olympiad 1997 - 98 (Russia), 11.11
An arbitrary point $M$ is taken on the basis of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. Prove that the straight line passing through the points $M$ and$ P$ intersects the height of the pyramid (or its extension). Let us denote this intersection point by $E$. Find $MP: PE$ if the dihedral angles at the base of the pyramid are equal to $a$.
1991 Arnold's Trivium, 100
Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection.
2014 ELMO Shortlist, 10
Find all positive integer bases $b \ge 9$ so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers $n$.
[i]Proposed by Yang Liu[/i]
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
1963 Polish MO Finals, 6
Through the vertex of a trihedral angle in which no edge is perpendicular to the opposite face, a straight line is drawn in the plane of each face perpendicular to the opposite edge. Prove that the three straight lines obtained lie in one plane.
2008 Oral Moscow Geometry Olympiad, 5
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov).
1993 All-Russian Olympiad Regional Round, 11.6
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
2015 Oral Moscow Geometry Olympiad, 5
A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.
1987 Austrian-Polish Competition, 1
Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.
1984 IMO Shortlist, 13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2012 District Olympiad, 4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$
2006 Polish MO Finals, 2
Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$. Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.
2001 AIME Problems, 15
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.
2007 AIME Problems, 5
The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
2017 Simon Marais Mathematical Competition, B1
Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex.
She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.
1957 AMC 12/AHSME, 38
From a two-digit number $ N$ we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:
$ \textbf{(A)}\ {N}\text{ cannot end in 5}\qquad\\
\textbf{(B)}\ {N}\text{ can end in any digit other than 5}\qquad \\
\textbf{(C)}\ {N}\text{ does not exist}\qquad \\
\textbf{(D)}\ \text{there are exactly 7 values for }{N}\qquad \\
\textbf{(E)}\ \text{there are exactly 10 values for }{N}$
2004 AIME Problems, 11
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
1990 Tournament Of Towns, (261) 5
Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to
(a) no less than $ 5$ faces?
(b) exactly $5$ faces?
(G. Galperin)
2025 Sharygin Geometry Olympiad, 24
The insphere of a tetrahedron $ABCD$ touches the faces $ABC$, $BCD$, $CDA$, $DAB$ at $D^{\prime}$, $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Denote by $S_{AB}$ the area of the triangle $AC^{\prime}B^{\prime}$. Define similarly $S_{AC}$, $S_{BC},$ $S_{AD}$, $S_{BD}$, $S_{CD}$. Prove that there exists a triangle with sidelengths $\sqrt{S_{AB}S_{CD}}$, $\sqrt{S_{AC}S_{BD}}$ , $\sqrt{S_{AD}S_{BC}}$.
Proposed by: S.Arutyunyan
1965 IMO, 3
Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.
2018 Costa Rica - Final Round, 6
The four faces of a right triangular pyramid are equilateral triangles whose edge measures $3$ dm. Suppose the pyramid is hollow, resting on one of its faces at a horizontal surface (see attached figure) and that there is $2$ dm$^3$ of water inside. Determine the height that the liquid reaches inside the pyramid.
[img]https://cdn.artofproblemsolving.com/attachments/0/7/6cd6e1077620371e56ed57d19fd3d05a58904e.png[/img]