Found problems: 2265
2019 Adygea Teachers' Geometry Olympiad, 3
In a cube-shaped box with an edge equal to $5$, there are two balls. The radius of one of the balls is $2$. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.
1980 Bulgaria National Olympiad, Problem 6
Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.
2005 Tuymaada Olympiad, 4
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.
[i]Proposed by L. Emelyanov[/i]
1967 IMO Longlists, 20
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
1976 IMO Longlists, 17
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.
2021 Belarusian National Olympiad, 9.2
A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once.
Prove that he made an even amount of moves that changed the face he is on.
2015 Bangladesh Mathematical Olympiad, 5
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
2020 AMC 10, 20
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
$\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38$
1986 IMO Longlists, 8
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
2012 Today's Calculation Of Integral, 797
In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$.
Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$.
50 points
I Soros Olympiad 1994-95 (Rus + Ukr), 11.10
Given a tetrahedron $A_1A_2A_3A_4$ (not necessarily regulart). We shall call a point $N$ in space [i]Serve point[/i], if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by $a (N)$ and call the [i]Serve plane[/i] of the point $N$. By $B_{ij}$ denote, respectively, the midpoint of the edges $A_1A_j$, $1\le i <j \le 4$. For each point $M$, denote by $M_{ij}$ the points symmetric to $M$ with respect to $B_{ij},$ $1\le i <j \le 4$. Prove that if all points $M_{ij}$ are Serve points, then the point $M$ belongs to all Serve planes $a (M_{ij})$, $1\le i <j \le 4$.
IV Soros Olympiad 1997 - 98 (Russia), 11.9
Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if:
a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right
b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$.
Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown
1954 Moscow Mathematical Olympiad, 284
How many planes of symmetry can a triangular pyramid have?
1968 Yugoslav Team Selection Test, Problem 6
Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.
2016 Sharygin Geometry Olympiad, P22
Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.
2018-2019 SDML (High School), 13
A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube?
[NEEDS DIAGRAM]
$ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$
2017 District Olympiad, 2
Let $ ABCDA’B’C’D’ $ a cube. $ M,P $ are the midpoints of $ AB, $ respectively, $ DD’. $
[b]a)[/b] Show that $ MP, A’C $ are perpendicular, but not coplanar.
[b]b)[/b] Calculate the distance between the lines above.
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$.
Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$.
When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$
2003 Polish MO Finals, 5
The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.
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.
1989 AMC 12/AHSME, 26
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$ \textbf{(A)}\ \frac{\sqrt{3}}{12} \qquad\textbf{(B)}\ \frac{\sqrt{6}}{16} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{8} \qquad\textbf{(E)}\ \frac{1}{4} $
1986 IMO Longlists, 12
Let $O$ be an interior point of a tetrahedron $A_1A_2A_3A_4$. Let $ S_1, S_2, S_3, S_4$ be spheres with centers $A_1,A_2,A_3,A_4$, respectively, and let $U, V$ be spheres with centers at $O$. Suppose that for $i, j = 1, 2, 3, 4, i \neq j$, the spheres $S_i$ and $S_j$ are tangent to each other at a point $B_{ij}$ lying on $A_iA_j$ . Suppose also that $U $ is tangent to all edges $A_iA_j$ and $V$ is tangent to the spheres $ S_1, S_2, S_3, S_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.
1986 Traian Lălescu, 1.4
Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression
$$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$
has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $
1993 Bulgaria National Olympiad, 3
it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with $7$ vertices) with common base $A_1A_2A_3A_4A_5A_6A_7$ and vertices respectively the points $B$ and $C$. The edges $BA_i , CA_i$ $(i = 1,...,7$), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.
2013 F = Ma, 11
A right-triangular wooden block of mass $M$ is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass $m$, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table?
$\textbf{(A) } 2mg\\
\textbf{(B) } 2mg + Mg\\
\textbf{(C) } mg + Mg\\
\textbf{(D) } Mg + mg( \sin \alpha + \sin \beta)\\
\textbf{(E) } Mg + mg( \cos \alpha + \cos \beta)$