A spacecraft landed on an asteroid. It is known that the asteroid is either a ball or a cube. The rover started its route at the landing site and finished it at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover transmitted its spatial coordinates to the spacecraft on the landing site so that the trajectory of the rover movement was known. Can it happen that this information is not suffcient to determine whether the asteroid is a ball or a cube?

A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments as shown below. This cup has height $h$ and volume $V$. A second cup that is exactly the same shape as the first is held upright inside the first cup so that its base is a distance of $\tfrac{h}2$ from the base of the first cup. The volume of liquid that will t inside the first cup and outside the second cup can be written $\tfrac{m}{n}\cdot V$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] pair s = (10,1); draw(ellipse((0,0),4,1)^^ellipse((0,-6),2,.5)); fill((3,-6)--(-3,-6)--(0,-2.1)--cycle,white); draw((4,0)--(2,-6)^^(-4,0)--(-2,-6)); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-2.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); pair s = (10,-2); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-4.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); //darn :([/asy]

A plane cuts a right circular cone of volume $ V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. [i]Original formulation:[/i] A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

4. Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space. If point $Q$ move to points on S expect $(0,0,2)$. Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$. Find the range of moving $R$, then illustrate it.

Four small spheres each with radius $6$ are each internally tangent to a large sphere with radius $17$. The four small spheres form a ring with each of the four spheres externally tangent to its two neighboring small spheres. A sixth intermediately sized sphere is internally tangent to the large sphere and externally tangent to each of the four small spheres. Its radius is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/2/25955cd6f22bc85f2f3c5ba8cd1ee0821c9d50.png[/img]

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$. (a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$. (b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.

A sphere of radius $\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron?

How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.

I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals $\textbf{(A) }\sqrt[7]{12}\qquad\textbf{(B) }2\sqrt[7]{12}\qquad\textbf{(C) }\sqrt[7]{32}\qquad\textbf{(D) }\sqrt[12]{32}\qquad\textbf{(E) }2\sqrt[12]{32}$

Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?