Found problems: 130
$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there?
Two fillings are the same if one filling can be transformed to the other filling via a translation.
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Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube
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.
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
Divide a cube into three equal pyramids.
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.
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?
The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .
A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the
top faces of all the cubes are the same colour) by means of such operations?
( D. Fomin , Leningrad)
Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.
An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.
Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.
Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, the shortest distance the ant must crawl to reach the opposite corner of the cube can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$.
A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.
* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.
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.
How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point.
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 $CC_1$.
Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube?
Note: Two colorings are considered the same if one can be obtained from the other by rotation.
Let $ABCDA'B'C'D'$ be the rectangular parallelepiped.
Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$.
a) Prove that the points $D, O, P$ are collinear.
b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.
A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?