Found problems: 130
Place the integers $1,2 , \ldots, n^{3}$ in the cells of a $n\times n \times n$ cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?
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) $\frac13 < x \leq\frac12$.
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.
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ .
A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.
Find the side length of the largest square that can be inscribed in the unit cube.
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]
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)
Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$
a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel.
b) Calculate the distance between these planes, knowing that $AB = 1$.
* What is the radius of the largest possible circle inscribed into a cube with side $a$?
In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that:
a) $MM_1 = MM_2$
b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel AD$;
c) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.
$C$ is a cube side $1$. The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?
Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.
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 $10^m<B(2015^{100})<10^{m+1}$.
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
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.
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.
A large cube of size \(4 \times 4 \times 4\) is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:
In each block of \(1 \times 1 \times 4\), \(1 \times 4 \times 1\), and \(4 \times 1 \times 1\) cubes, there must be exactly one red cube.
Determine how many different ways it is possible to choose the 16 small cubes to be colored red.
Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.
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 $M$?
Compute the radius of the largest circle that fits entirely within a unit cube.
[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[i]Alexandru Sergiu Alamă[/i]
It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?
A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick .
(A . Andjans , Riga)
Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon