Found problems: 473
A sphere is inscribed in the tetrahedron whose vertices are $A=(6,0,0), B=(0,4,0), C=(0,0,2),$ and $D=(0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.
Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying
\[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\]
in $xyz$-space.
(1) Find $V(r)$.
(2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$
(3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$
Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$
This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.
Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.
A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere.
A. Zaslavsky
A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
$ \textbf{(A)}\; 3+\frac{\sqrt{30}}2\qquad
\textbf{(B)}\; 3+\frac{\sqrt{69}}3\qquad
\textbf{(C)}\; 3+\frac{\sqrt{123}}4\qquad
\textbf{(D)}\; \frac{52}9\qquad
\textbf{(E)}\; 3+2\sqrt{2} $
Let $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that
\[ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4\]
and
\[ \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.\]
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.
Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.
A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$
[b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$
The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.
Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.
Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$.
[i]Stere Ianuș[/i]
$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?
A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 4\sqrt5 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 6\sqrt3$
Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.
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.
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)
$ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $