Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality \[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\] holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals. a) Prove that $BCPN$ is also a cyclic quadrilateral. b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.

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 $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent. [i](7 points)[/i]

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

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 $p$.

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$

Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls.

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.

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions: i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$; ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

In a dihedral angle of measure $c$ two non-intersecting spheres are inscribed, the centers of which are located on a straight line perpendicular to the edge of the dihedral angle. The points of contact of these spheres with the edges of the corner are at distances $a$ and $b$ from the edge. Let us consider an arbitrary plane tangent to these spheres and intersecting the segment connecting their centers. Let us denote by $\phi$ the measure of the angle formed at the intersection of this plane with the faces of a given dihedral angle. Find the greatest value $\phi$.

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 $m+n.$

The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.

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]

Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.

Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.

We say that a rational number is [i]spheric[/i] if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that: [b]a)[/b] $ 7 $ is not spheric. [b]b)[/b] a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest. [asy] size(225); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.7)); draw((0,-1)--(2,-1),EndArrow); label("$\vec{F}$",(1, -1),S); label("Disk",(-1,0),W); filldraw(circle((5,0),1),gray(.7)); filldraw(circle((5,0),0.75),white); draw((5,-1)--(7,-1),EndArrow); label("$\vec{F}$",(6, -1),S); label("Hoop",(6,0),E); filldraw(circle((10,0),1),gray(.5)); draw((10,-1)--(12,-1),EndArrow); label("$\vec{F}$",(11, -1),S); label("Sphere",(11,0),E); [/asy] $ \textbf{(A)} \ \text{disk, hoop, sphere}$ $\textbf{(B)}\ \text{sphere, disk, hoop}$ $\textbf{(C)}\ \text{hoop, sphere, disk}$ $\textbf{(D)}\ \text{disk, sphere, hoop}$ $\textbf{(E)}\ \text{hoop, disk, sphere} $

Four different points $A,B,C,D$ are given in space such that $AC\perp BD,AD\perp BC.$ Show there is a sphere containing midpoits of all 7 segments $AB,AC,AD,BC,BD,CD.$

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.

Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular. For $R$ the maximum possible radius of this sphere, find $R^2$.

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.