What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon? (A Shapovalov)

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?

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

A cube with the edge of length $n$ is divided onto $n^3$ unit ones. Let us choose some of them and draw three lines parallel to the edges through their centres. What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes? a) Find the answer for the small $n$ ($n = 2,3,4$). b) Try to find the answer for $n = 10$. c) If You can not solve the general problem, try to estimate that value from the upper and lower side. d) Note, that You can reformulate the problem in such a way: Consider all the triples $(x_1,x_2,x_3)$, where $x_i$ can be one of the integers $1,2,...,n$. What is the minimal number of the triples necessary to provide the property: [i]for each of the triples there exist the chosen one, that differs only in one coordinate. [/i] Try to find the answer for the situation with more than three coordinates, for example, with four.

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

A sphere sits inside a cubic box, tangent on all $6$ sides of the box. If a side of the box is $5$, and the volume of the sphere is $x\pi$ , find $x$.

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

Anton the ant takes a walk along the vertices of a cube. He starts at a vertex and stops when it reaches this point again. Between two vertices it moves over an edge, a side face diagonal or a space diagonal. During the rout it visits each of the other vertices exactly [i]once [/i] and nowhere intersects its road already traveled. (a) Show that Anton walks along at least one edge. (b) Show that Anton walks along at least two edges.

Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$. a) Prove that $BM + BN + BP = BD'$. b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.

Let $VWXYZ$ be a square pyramid with vertex $V$ with height $1$, and with the unit square as its base. Let $STANFURD$ be a cube, such that face $FURD$ lies in the same plane as and shares the same center as square face $WXYZ$. Furthermore, all sides of $FURD$ are parallel to the sides of $WXY Z$. Cube $STANFURD$ has side length $s$ such that the volume that lies inside the cube but outside the square pyramid is equal to the volume that lies inside the square pyramid but outside the cube. What is the value of $s$?

How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.

Let $A$ be a corner of a cube. Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below: [center][img]http://i.imgur.com/tEODnV0.png[/img][/center] The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$. [list=a] [*] How many sides does $P$ have? Justify your answer. [*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this. For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$. Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.

A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.

The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?

Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$. a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$. b) Calculate the distance from $D'$ to the line $EF$.