Olympiad problems

1988 Tournament Of Towns, (187) 4

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.

1995 Romania Team Selection Test, 2

Tags: parallelepiped , combinatorial geometry , combinatorics , volume , cube , partition

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.

1995 Chile National Olympiad, 4

Tags: cube , combinatorics , digit

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?

2013 Tournament of Towns, 5

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

A spacecraft landed on an asteroid. It is known that the asteroid is either a ball or a cube. The rover started its route at the landing site and finished it at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover transmitted its spatial coordinates to the spacecraft on the landing site so that the trajectory of the rover movement was known. Can it happen that this information is not suffcient to determine whether the asteroid is a ball or a cube?

1981 All Soviet Union Mathematical Olympiad, 321

Tags: combinatorics , cube , geometry , 3d geometry

A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:

1962 IMO, 3

Tags: geometry , 3d geometry , perimeter , vector , cube

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 $XY$.

1977 Czech and Slovak Olympiad III A, 1

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

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.

2012 Polish MO Finals, 2

Tags: 3d geometry , cube , geometry , combinatorics

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.

2014 Romania National Olympiad, 2

Tags: geometry , 3d geometry , cube , angle , distance

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 $EF$.

Kyiv City MO 1984-93 - geometry, 1993.11.3

Tags: geometry , 3d geometry , sphere , cube

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

1975 All Soviet Union Mathematical Olympiad, 216

Tags: geometry , cube , coloring

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.)

2022 Olimphíada, 1

Tags: number theory , cube

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.

1963 Dutch Mathematical Olympiad, 5

Tags: combinatorics , coloring , cube

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?

1969 IMO Shortlist, 39

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

$(HUN 6)$ 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.

1957 Polish MO Finals, 6

Tags: geometry , 3d geometry , cube

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.

2021 AIME Problems, 6

Tags: 3d geometry , analytic geometry , cube

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 $AP.$

1966 IMO Shortlist, 57

Tags: geometry , 3d geometry , symmetry , rotation , cube

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.

1985 Tournament Of Towns, (093) 1

Tags: projection , cube , perpendicular , plane , 3d geometry , geometry

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.

1976 All Soviet Union Mathematical Olympiad, 224

Tags: combinatorics , cube , 3-digit

Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?

2012 Sharygin Geometry Olympiad, 1

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

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)

1977 Czech and Slovak Olympiad III A, 6

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

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.

2018 Hanoi Open Mathematics Competitions, 1

Tags: geometry , rectangle , cube , 3d geometry

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. $16$

1966 Polish MO Finals, 3

Tags: geometry , 3d geometry , parallelepiped , 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.

1959 AMC 12/AHSME, 1

Tags: geometry , 3d geometry , percent , percent increase , cube

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 $

2018 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , cube , angle , plane

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 $BB_1C_1$.