Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior $x^2 +y^2 +z^2 +t^2 <r^2$

Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that (a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and (b) the Euclidean norm of every point $v$ in $A$ satisfies \[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\] Prove that the cardinality of $A$ is at most $2d+4$.

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} $

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 $

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

Watermelon(a sphere) with radius $R$ lies on a table. $n$ flies fly above the table, each at distance $\sqrt{2}R$ from the center of the watermelon. At some moment any fly couldn't see any of the other flies. (Flies can't see each other, if the segment connecting them intersects or touches watermelon). Find the maximum possible value of $n$

$n$ 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.

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$.

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.

On a table are a cone, resting on the base, and six equal spheres tangent to the cone. Besides that, each sphere is tangent to the two adjacent spheres. Knowing that the radius $R$ of the base of the cone is half its height and determine the radius $r$ of the spheres.

A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$.

(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines. (b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.

Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.

The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$

Some identically sized spheres are piled in $n$ layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of $n^2$ spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer $m$ such that as $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sqrt{m}}{\pi}$. Find $m$. [center][img]https://snag.gy/bIwyl6.jpg[/img][/center]

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

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]

The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.

A rectangular box $ P$ is inscribed in a sphere of radius $ r$. The surface area of $ P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $ r$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

$(GBR 3)$ 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 $\sin \phi = \frac{r \tan \alpha}{h}$.