Found problems: 2265
Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$.
[img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]
Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .
Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$
At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .
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]
In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.
Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.
Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.
Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.
Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).
A tetrahedron has three edges of length $2$ and three edges of length $4$, and one of its faces is an equilateral triangle. Compute the radius of the sphere that is tangent to every edge of this tetrahedron.
In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.
Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$
In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$.
[asy]
import three;
size(250);
defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2");
currentprojection = orthographic(0,-15,5);
draw(circle((0,0,0), 15),dashes);
draw(circle((0,0,80), 15));
draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0)));
draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80));
draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes);
draw("48", (-24,0,-20)--(24,0,-20));
draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17));
label("30", (0,0,-15));
draw("60", (50,0,0)--(50,0,60));
draw("20", (50,0,60)--(50,0,80));
draw((50,0,60)--(47,0,60));[/asy]
Suppose that the closed subset $K$ of the sphere
$$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$
is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between
$$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$
and $K$ is less than $\varepsilon$.
Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.
A cylindrical glass beaker is $8$ cm high and its circumference rim is $12$ cm wide . Inside, $3$ cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located $1$ cm from the base (or bottom) of the glass, there is a fly.
What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path?
[hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca.
¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?[/hide]
Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.
A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.
In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$.
Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.
Points $STABCD$ in space form a convex octahedron with faces $SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA$ such that there exists a sphere that is tangent to all of its edges. Prove that $A,B,C,D$ lie in one plane.
Is there a tetrahedron $ABCD$ such that $AB+BC+CD+DA=12\text{ cm}$ with volume $\mathrm V\ge2\sqrt3\text{ cm}^3?$