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.

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Found problems: 130

2013 German National Olympiad, 4
Tags: geometry , 3d geometry , cube , length , path
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 $a.$

2018 Romania National Olympiad, 4
Tags: geometry , 3d geometry , cube , parallel , equal segments , angle
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) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.

1996 Romania National Olympiad, 3
Tags: 3d geometry , cube , parallelepiped , 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.

1988 Tournament Of Towns, (187) 4
Tags: coloring , combinatorics , cube , geometry , 3d geometry
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.

1979 Chisinau City MO, 182
Tags: geometry , cube , plane , pentagon , 3d geometry
Prove that a section of a cube by a plane cannot be a regular pentagon.

1969 IMO Longlists, 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.

2004 Junior Balkan Team Selection Tests - Romania, 4
Tags: combinatorics , cube
Consider a cube and let$ M, N$ be two of its vertices. Assign the number $1$ to these vertices and $0$ to the other six vertices. We are allowed to select a vertex and to increase with a unit the numbers assigned to the $3$ adjiacent vertices - call this a [i]movement[/i]. Prove that there is a sequence of [i]movements [/i] after which all the numbers assigned to the vertices of the cube became equal if and only if $MN$ is not a diagonal of a face of the cube. Marius Ghergu, Dinu Serbanescu

2018 District Olympiad, 3
Tags: geometry , 3d geometry , cube , parallelepiped , collinear
Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

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?

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 $

2005 iTest, 12
Tags: geometry , 3d geometry , sphere , cube
A sphere sits inside a cubic box, tangent on all $6$ sides of the box. If a side of the box is $5$, and the volume of the sphere is $x\pi$ , find $x$.

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?

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

2000 Austrian-Polish Competition, 6
Tags: cube , 3d geometry , geometry , combinatorics
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.

1960 IMO Shortlist, 5
Tags: geometry , 3d geometry , cube , locus , locus problems
Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

2002 District Olympiad, 4
Tags: perimeter , 3d geometry , cube , geometric inequality , geometry
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 $2a\sqrt3$.

IV Soros Olympiad 1997 - 98 (Russia), 10.11
Tags: geometry , 3d geometry , cube , plane , geometric inequality
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?

Denmark (Mohr) - geometry, 1996.3
Tags: geometry , 3d geometry , sphere , ratio , cube
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.

1993 Tournament Of Towns, (361) 4
Tags: combinatorics , cube , combinatorial geometry , geometry , 3d geometry
An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

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.

2014 BMT Spring, 7
Tags: geometry , 3d geometry , pyramid , cube
Let $VWXYZ$ be a square pyramid with vertex $V$ with height $1$, and with the unit square as its base. Let $STANFURD$ be a cube, such that face $FURD$ lies in the same plane as and shares the same center as square face $WXYZ$. Furthermore, all sides of $FURD$ are parallel to the sides of $WXY Z$. Cube $STANFURD$ has side length $s$ such that the volume that lies inside the cube but outside the square pyramid is equal to the volume that lies inside the square pyramid but outside the cube. What is the value of $s$?

1984 All Soviet Union Mathematical Olympiad, 394
Tags: combinatorial geometry , geometry , area , 3d geometry , cube
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

2015 BMT Spring, 9
Tags: 3d geometry , cube , geometric inequality , square , geometry
Find the side length of the largest square that can be inscribed in the unit cube.

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?

1952 Moscow Mathematical Olympiad, 228
Tags: geometry , 3d geometry , cylinder , cube
How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.