Found problems: 473
The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$.
[i]2022 CCA Math Bonanza Team Round #3[/i]
Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.
For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.)
[i]Z. Szabo[/i]
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.
Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.
$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.
A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB\equal{}CD$, $ AC\equal{}BD$ and $ AD\equal{}BC$.
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$
$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.
Eight spheres with radius of $1$ are put into a circular column. There are two floors, and each sphere is tangent to adjacent four spheres, one of the bottom surfaces, and the flank. Then the height of the circular column is________.
Consider a solid hemisphere of radius $1$. Find the distance from its center of mass to the base.
Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.
A sphere with unit radius is given. Furthermore, circles $k_0,k_1,\ldots,k_n\ (n\ge3)$ of the same radius $r$ are given on the sphere. The circle $k_0$ is tangent to all other circles $k_i$ and every two circles $k_i,k_{i+1}$ are tangent for $i=1,\ldots,n$ (assuming $k_{n+1}=k_1$).
a) Find relation between numbers $n,r.$
b) Determine for which $n$ the described situation can occur and compute the corresponding radius $r.$
(We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
[b](a)[/b] [Problem for class 11]
Let r be the inradius and $r_a$, $r_b$, $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$.
[b](b)[/b] [Problem for classes 12/13]
Let r be the radius of the insphere and let $r_a$, $r_b$, $r_c$, $r_d$ the radii of the four exspheres of a tetrahedron ABCD. (An [i]exsphere[/i] of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.)
Prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$.
I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;)
Darij
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 $\angle ADD'=\angle BDD'$.
A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
$$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$
is the sphere
$$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$