This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 130

2020 Yasinsky Geometry Olympiad, 6
Tags: geometry , 3d geometry , cube , plane geometry , area
A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases. (Alexander Shkolny)

1948 Moscow Mathematical Olympiad, 153
Tags: maximum , radical , cube , 3d geometry , combinatorial geometry , circles , geometry
* What is the radius of the largest possible circle inscribed into a cube with side $a$?

1935 Moscow Mathematical Olympiad, 014
Tags: geometry , locus , locus in space , cube
Find the locus of points on the surface of a cube that serve as the vertex of the smallest angle that subtends the diagonal.

1974 Vietnam National Olympiad, 4
Tags: cube , 3d geometry , geometry
$C$ 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?

2013 German National Olympiad, 4
Tags: path , geometry , 3d geometry , cube , length
Let $ABCDEFGH$ be a cube of sidelength $a$ and such that $AG$ is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points $P$ on the surface for which the shortest path from $P$ to $A$ and from $P$ to $G$ have the same length $l.$ Also determine all possible values of $l$ depending on $a.$

1996 Bundeswettbewerb Mathematik, 1
Tags: combinatorial geometry , geometry , combinatorics , cube
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?

Kyiv City MO 1984-93 - geometry, 1985.10.2
Tags: 3d geometry , cube , angle , geometry
Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?

1993 ITAMO, 6
Tags: rotation , geometry , 3d geometry , volume , cube
A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.

1988 All Soviet Union Mathematical Olympiad, 481
Tags: geometry , cube , 3d geometry , broken line , minimum
A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

1990 ITAMO, 1
Tags: combinatorics , cube , combinatorial geometry , geometry , 3d geometry
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.

1995 All-Russian Olympiad Regional Round, 11.2
Tags: 3d geometry , cube , parallelepiped , geometry
A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

1994 ITAMO, 5
Tags: area , minimum , maximum , cube , plane , geometry
Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

2016 Romania National Olympiad, 2
Tags: 3d geometry , geometry , cube , perpendicular , ratio
In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$

Indonesia Regional MO OSP SMA - geometry, 2003.3
Tags: geometry , rectangle , 3d geometry , cube
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 $DPFQ$ .

1985 All Soviet Union Mathematical Olympiad, 411
Tags: coloring , combinatorics , combinatorial geometry , cube , parallelepiped , geometry
The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?

1997 Poland - Second Round, 6
Tags: point , geometry , 3d geometry , combinatorial geometry , cube , distance , max
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

2007 Oral Moscow Geometry Olympiad, 1
Tags: geometry , 3d geometry , cube
Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut). (V. Shevyakov)

1988 Tournament Of Towns, (188) 1
Tags: combinatorics , cube
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)

2002 District Olympiad, 4
Tags: perimeter , geometry , 3d geometry , cube , geometric inequality
The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L \in [CC' ], M \in [D'A']$. a) Show that $\sqrt3 KL \ge KB + BC + CL$ b) Show that the perimeter of triangle $KLM$ is strictly greater than $2a\sqrt3$.

2017 Yasinsky Geometry Olympiad, 5
Tags: 3d geometry , 3-dimensional geometry , cube , area , geometry
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 $CC_1$.

1954 Czech and Slovak Olympiad III A, 4
Tags: 3d geometry , ratio , locus , cube , 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.

1996 Denmark MO - Mohr Contest, 3
Tags: 3d geometry , sphere , ratio , cube , 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.

May Olympiad L2 - geometry, 2003.5
Tags: geometry , 3d geometry , cube , minimum
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.

2014 BAMO, 1
Tags: cube , geometry , 3d geometry , counting , combinatorics , arrangement
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.

1953 Moscow Mathematical Olympiad, 255
Tags: geometry , 3d geometry , pyramid , cube
Divide a cube into three equal pyramids.