Olympiad problems

1967 IMO Shortlist, 3

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

1997 Tournament Of Towns, (555) 5

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

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)

1954 Czech and Slovak Olympiad III A, 4

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

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.

IV Soros Olympiad 1997 - 98 (Russia), 10.11

Tags: 3d geometry , plane , cube , geometric inequality , geometry

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?

1963 Dutch Mathematical Olympiad, 5

Tags: coloring , cube , combinatorics

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

$(HUN 6)$ 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.

Indonesia Regional MO OSP SMA - geometry, 2003.3

Tags: 3d geometry , rectangle , cube , geometry

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 $DPFQ$ .

2015 Czech-Polish-Slovak Junior Match, 6

Tags: sum , product , cube , combinatorics

The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.

2018 Adygea Teachers' Geometry Olympiad, 4

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

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 $BB_1C_1$.

1966 IMO Shortlist, 57

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

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.

2002 Estonia National Olympiad, 4

Tags: max , cube , 3d geometry , line , geometry

Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.

1976 All Soviet Union Mathematical Olympiad, 224

Tags: 3-digit , cube , combinatorics

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?

1989 Austrian-Polish Competition, 5

Tags: 3d geometry , cube , product , sphere , maximum , geometry

Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$. We consider lines $g$ through $A$ containing at least one point of $k$. Let $P$ be the intersection point of $g$ and $k$ closer to $A$, and $Q$ be the second intersection point of $g$ and $\omega$. Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.

2015 BMT Spring, 9

Tags: geometric inequality , 3d geometry , cube , geometry , square

Find the side length of the largest square that can be inscribed in the unit cube.

1984 All Soviet Union Mathematical Olympiad, 376

Tags: cube , geometry , coloring , game strategy , combinatorics

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?

1962 IMO, 3

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

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 $XY$.

1935 Moscow Mathematical Olympiad, 019

Tags: dodecahedron , cube , coloring , combinatorics

a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)

1996 Bundeswettbewerb Mathematik, 1

Tags: cube , geometry , combinatorics , combinatorial geometry

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?

1985 All Soviet Union Mathematical Olympiad, 417

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

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 $B_1$ .

2013 Tournament of Towns, 5

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

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?

1995 All-Russian Olympiad Regional Round, 10.7

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

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

1996 Romania National Olympiad, 3

Tags: parallelepiped , 3d geometry , cube , geometry

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.

2012 Polish MO Finals, 2

Tags: 3d geometry , cube , combinatorics , geometry

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 $K$.

1996 Denmark MO - Mohr Contest, 3

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

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.

2021 AIME Problems, 6

Tags: cube , 3d geometry , analytic geometry

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 $AP.$