Found problems: 2265
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
$ \textbf{(A)}\ \frac {5}{256} \qquad
\textbf{(B)}\ \frac {21}{1024} \qquad
\textbf{(C)}\ \frac {11}{512} \qquad
\textbf{(D)}\ \frac {23}{1024} \qquad
\textbf{(E)}\ \frac {3}{128}$
Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
A variable tetrahedron $ABCD$ has the following properties:
Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?
Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property.
(b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.
What is the largest number of parts into which $n$ planes can divide space?
We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).
Show that there do not exist more than $27$ half-lines (or rays) emanating from the origin in the $3$-dimensional space, such that the angle between each pair of rays is $\geq \frac{\pi}{4}$.
Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure.
Find the area of intersection of the cube with the plane through the points $a,m,e$.
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]
* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.
In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$)
Prove that faces of this tetrahedron are four congruent triangles.
This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.
[img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]
How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the above}
$
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$ where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $.
(Alexey Klurman)
Show how to divide space into
(a) congruent tetrahedra,
(b) congruent “equifaced” tetrahedra.
(A tetrahedron is called equifaced if all its faces are congruent triangles.)
(NB Vassiliev)
We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$.
a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8;
b) Prove that any prism with 2000 vertices is [i]binary[/i].
Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.
The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$, where $m$, $n$, and $p$ are integers. Find $m+n+p$.
Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.