Found problems: 130
Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$).
a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$;
b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.
Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same?
(A Shapovalov)
A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.
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.
A $3\times 3\times 3$ cube is made of $1\times 1\times 1$ cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features:
1) Projection of this solid on each face of the original cube is a $3\times 3$ square,
2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).
One of the numbers $1$ or $-1$ is assigned to each vertex of a cube. To each face of the cube is assigned the integer which is the product of the four integers at the vertices of the face. Is it possible that the sum of the $14$ assigned integers is $0$?
(G. Galperin)
A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i].
Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].
Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.
Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment).
(A.Shapovalov)
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 $EF$.
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.
Each vertex of a cube is assigned $1$ or $-1$. Each face is assigned the product of the four numbers at its vertices. Determine all possible values that can be obtained as the sum of all the $14$ assigned numbers.
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is:
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $
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.
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.
Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.
Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$
Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$?
(V Proizvolov)
Let $n$ and $p$ be positive integers, with $p>3$ prime, such that:
i) $n\mid p-3;$
ii) $p\mid (n+1)^3-1.$
Show that $pn+1$ is the cube of an integer.
You want to color the side faces of a cube in such a way that each face is colored evenly. Six colors are available:
[i]red, white, blue, yellow, purple, orange[/i]. Two cube colors are called the same if one arises from the other by a rotation of the cube.
(a) How many different cube colorings are there, using six colors?
(b) How many different cube colorings are there, using exactly five colors?
A nonnegative real number is written at every cube's vertex. The sum of those numbers equals to $1$. Two players choose in turn faces of the cube, but they cannot choose the face parallel to already chosen one (the first moves twice, the second -- once). Prove that the first player can provide the number, at the common for three chosen faces vertex, to be not greater than $1/6$.
Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?