Found problems: 473
Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?
A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.
KöMaL A.723. (April 2018), G. Kós
The surface areas of the bases of a given truncated triangular pyramid are equal to $ B_1 $ and $ B_2 $. This pyramid can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. Prove that the lateral surface area of the given pyramid is $ (\sqrt{B_1} + \sqrt{B_2})(\sqrt[4]{B_1} + \sqrt[4]{B_2})^2 $.
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at $A$, $B$, and $C$. Compute the area of triangle $ABC$.
[i]2021 CCA Math Bonanza Team Round #6[/i]
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 $\sqrt{3} :\sqrt{3} :2$.
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.
Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior
$x^2 +y^2 +z^2 +t^2 <r^2$
Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.
Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$. We consider lines $g$ through $A$ containing at least one point of $k$. Let $P$ be the intersection point of $g$ and $k$ closer to $A$, and $Q$ be the second intersection point of $g$ and $\omega$. Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.
Through points $ A $ and $ B $ two oblique lines $ m $ and $ n $ are drawn perpendicular to the line $ AB $. On line $ m $ the point $ C $ (different from $ A $) is taken, and on line $ n $ the point $ D $ (different from $ B $) is taken. Given the lengths of segments $ AB = d $ and $ CD = l $ and the angle $ \varphi $ formed by the oblique lines $ m $ and $ n $, calculate the radius of the surface of the sphere passing through the points $ A $, $ B $, $ C $, $ D $.
$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.
A charged particle with charge $q$ and mass $m$ is given an initial kinetic energy $K_0$ at the middle of a uniformly charged spherical region of total charge $Q$ and radius $R$. $q$ and $Q$ have opposite signs. The spherically charged region is not free to move. Throughout this problem consider electrostatic forces only.
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
size(100);
filldraw(circle((0,0),1),gray(.8));
draw((0,0)--(0.5,sqrt(3)/2),EndArrow);
label("$R$",(0.25,sqrt(3)/4),SE);
[/asy]
(a) Find the value of $K_0$ such that the particle will just reach the boundary of the spherically charged region.
(b) How much time does it take for the particle to reach the boundary of the region if it starts with the kinetic energy $K_0$ found in part (a)?
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.
From a point in space, $n$ rays are issuing, whereas the angle among any two of these rays is at least $30^{\circ}$. Prove that $n < 59$.
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?
Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.
Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?
Define a $\emph{big circle}$ in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose $(AB)$ as the big circle that passes through $A$ and $B$. Also, let a $\emph{Spheric Triangle}$ be $3$ connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also $\angle XYZ$ the angle between $(XY)$ and $(YZ)$. Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S.
Let $\Delta ABC$ be a spheric triangle with all its angles $<90^{\circ}$ such that there is a circle $\omega$ tangent to $(BC)$,$(CA)$,$(AB)$ in $D,E,F$. Show that there is $P\in S$ with $\angle PAB=\angle DAC$, $\angle PCA=\angle FCB$, $\angle PBA=\angle EBC$.
Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.
A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?
[asy]
import graph3;
import solids;
real h=2+2*sqrt(7);
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2));
currentlight=light(4,-4,4);
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0));
draw(shift((1,3,1))*unitsphere,gray(0.85));
draw(shift((3,3,1))*unitsphere,gray(0.85));
draw(shift((3,1,1))*unitsphere,gray(0.85));
draw(shift((1,1,1))*unitsphere,gray(0.85));
draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85));
draw(shift((1,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,1,h-1))*unitsphere,gray(0.85));
draw(shift((1,1,h-1))*unitsphere,gray(0.85));
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h));
[/asy]
$\textbf{(A) }2+2\sqrt 7\qquad
\textbf{(B) }3+2\sqrt 5\qquad
\textbf{(C) }4+2\sqrt 7\qquad
\textbf{(D) }4\sqrt 5\qquad
\textbf{(E) }4\sqrt 7\qquad$