On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

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?

A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

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]

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

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

Given a sphere, a great circle of the sphere is a circle on the sphere whose diameter is also a diameter of the sphere. For a given positive integer $n,$ the surface of a sphere is divided into several regions by $n$ great circles, and each region is colored black or white. We say that a coloring is good if any two adjacent regions (that share an arc as boundary, not just a finite number of points) have different colors. Find, with proof, all positive integers $n$ such that in every good coloring with $n$ great circles, the sum of the areas of the black regions is equal to the sum of the areas of the white regions.

There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is $1$ sphere in the top row (which we will call row $0$), $3$ spheres in row $1$, $6$ spheres in row $2$, $10$ spheres in row $3$, etc. The top-most sphere in row $0$ is assigned the number $1$. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of $n$ for the sum of the numbers assigned to each sphere from row $0$ to row $n$. [asy] import three; import solids; size(8cm); //currentprojection = perspective(1, 1, 10); triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3); draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white); draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white); draw(surface(sphere(1,20)), white); label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20)); dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8); dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2); dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]

Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

A cube is divided into $27$ unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these $8$ spheres. What is the smallest possible value for the radius of the last sphere?

Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.

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]

In a cube-shaped box with an edge equal to $5$, there are two balls. The radius of one of the balls is $2$. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.

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.

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?

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

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.

A right rectangular prism is inscribed within a sphere. The total area of all the faces [of] the prism is $88$, and the total length of all its edges is $48$. What is the surface area of the sphere? $\text{(A) }40\pi\qquad\text{(B) }32\pi\sqrt{2}\qquad\text{(C) }48\pi\qquad\text{(D) }32\pi\sqrt{3}\qquad\text{(E) }56\pi$

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

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?

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.)