Found problems: 473
There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?
Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.
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 $S$?
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$?
$
\textbf{(A) }105 \qquad
\textbf{(B) }112\frac{1}{2} \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }135 \qquad
\textbf{(E) }150 \qquad
$
Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
What fraction of the Earth's volume lies above the $45$ degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.
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.
A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.
Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.
$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.
$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.
Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$
What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.
In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.
Calculate the mean value of the solid angle by which the disc $x^2 + y^2 \le 1$ lying in the plane $z = 0$ is seen from points of the sphere $x^2 + y^2 + (z-2)^2 = 1$.
Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.
Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?
Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$ be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$. .Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.
A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.)
Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.
Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent.
[hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]
Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?