Olympiad problems

1975 All Soviet Union Mathematical Olympiad, 216

For what $k$ is it possible to construct a cube $k\times k\times k$ of the black and white cubes $1\times 1\times 1$ in such a way that every small cube has the same colour, that have exactly two his neighbours. (Two cubes are neighbours, if they have the common face.)

1935 Moscow Mathematical Olympiad, 014

Tags: cube , locus in space , locus , geometry

Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

1988 Tournament Of Towns, (188) 1

Tags: cube , combinatorics

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)

2011 Sharygin Geometry Olympiad, 8

Tags: cube , combinatorial geometry , geometry

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?

1984 All Soviet Union Mathematical Olympiad, 394

Tags: combinatorial geometry , area , 3d geometry , cube , geometry

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

2014 BAMO, 1

Tags: arrangement , cube , 3d geometry , counting , combinatorics , geometry

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.

1997 Portugal MO, 2

Tags: cube , 3d geometry , geometry , isosceles

Consider the cube $ABCDEFGH$ and denote by, respectively, $M$ and $N$ the midpoints of $[AB]$ and $[CD]$. Let $P$ be a point on the line defined by $[AE]$ and $Q$ the point of intersection of the lines defined by $[PM]$ and $[BF]$. Prove that the triangle $[PQN]$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/0/0/57559efbad87903d087c738df279b055b4aefd.png[/img]

1993 Abels Math Contest (Norwegian MO), 4

Tags: cube , combinatorial geometry , sum , product , combinatorics

Each of the $8$ vertices of a given cube is given a value $1$ or $-1$. Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?

1994 Poland - Second Round, 4

Tags: cube , combinatorial geometry , combinatorics

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.

1987 Tournament Of Towns, (140) 5

Tags: combinatorics , coloring , cube , combinatorial geometry , rectangle

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)

1967 IMO Longlists, 27

Tags: cube , 3d geometry , polygon , intersection , geometry

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2014 Tournament of Towns., 6

Tags: cube , 3d geometry , projection , combinatorial geometry , geometry , combinatorics

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).

2012 Tournament of Towns, 4

Tags: cube , 3d geometry , sum , geometry

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.

2018 Hanoi Open Mathematics Competitions, 1

Tags: cube , 3d geometry , geometry , rectangle

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. $16$

2012 Sharygin Geometry Olympiad, 1

Tags: cube , square grid , 3d geometry , grid , geometry , rectangle

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)

1988 Tournament Of Towns, (182) 5

Tags: combinatorics , cube , combinatorial geometry

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)

1977 Czech and Slovak Olympiad III A, 6

Tags: cube , 3d geometry , volume , geometry , tetrahedron

A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.

1953 Moscow Mathematical Olympiad, 255

Tags: pyramid , 3d geometry , cube , geometry

Divide a cube into three equal pyramids.

2004 Nicolae Păun, 3

Tags: cube , 3d geometry , locus , geometry , sphere

[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]