Olympiad problems

1994 Poland - Second Round, 4

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.

2024 Regional Olympiad of Mexico Southeast, 3

Tags: combinatorics , cube , coloring

A large cube of size $4 \times 4 \times 4$ is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition: In each block of $1 \times 1 \times 4$, $1 \times 4 \times 1$, and $4 \times 1 \times 1$ cubes, there must be exactly one red cube. Determine how many different ways it is possible to choose the 16 small cubes to be colored red. Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.

1988 Tournament Of Towns, (182) 5

Tags: combinatorics , combinatorial geometry , cube

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)

2015 Czech-Polish-Slovak Junior Match, 6

Tags: sum , product , combinatorics , cube

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.

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?

1997 Portugal MO, 2

Tags: geometry , 3d geometry , isosceles , cube

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]

2017 Bundeswettbewerb Mathematik, 4

Tags: number theory , square , cube

We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.

1989 Austrian-Polish Competition, 5

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

1982 All Soviet Union Mathematical Olympiad, 330

Tags: cube , game strategy , combinatorics , geometry , 3d geometry

A nonnegative real number is written at every cube's vertex. The sum of those numbers equals to $1$. Two players choose in turn faces of the cube, but they cannot choose the face parallel to already chosen one (the first moves twice, the second -- once). Prove that the first player can provide the number, at the common for three chosen faces vertex, to be not greater than $1/6$.

2013 Flanders Math Olympiad, 3

Tags: combinatorics , combinatorial geometry , cube

Anton the ant takes a walk along the vertices of a cube. He starts at a vertex and stops when it reaches this point again. Between two vertices it moves over an edge, a side face diagonal or a space diagonal. During the rout it visits each of the other vertices exactly [i]once [/i] and nowhere intersects its road already traveled. (a) Show that Anton walks along at least one edge. (b) Show that Anton walks along at least two edges.

1992 ITAMO, 1

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

A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

2011 Sharygin Geometry Olympiad, 8

Tags: combinatorial geometry , cube , 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?

1995 All-Russian Olympiad Regional Round, 10.7

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

$N^3$ 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?

2015 BAMO, 4

Tags: geometry , 3d geometry , cube

Let $A$ be a corner of a cube. Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below: [center][img]http://i.imgur.com/tEODnV0.png[/img][/center] The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$. [list=a] [*] How many sides does $P$ have? Justify your answer. [*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.

2000 Tournament Of Towns, 4

Tags: cube , divisibility , combinatorics

Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property? (A Shapovalov)

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 $s$?

1935 Moscow Mathematical Olympiad, 019

Tags: dodecahedron , coloring , cube , 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.)

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.

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) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.

Kyiv City MO 1984-93 - geometry, 1990.11.1

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

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

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.

2014 Tournament of Towns., 6

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

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).