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: 67

2022 Vietnam TST, 2
Tags: polyhedron , combinatorics
Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number $26; 4$ and $2022$ (each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.

1988 Tournament Of Towns, (196) 3
Tags: polyhedron , combinatorics , combinatorial geometry
Prove that for each vertex of a polyhedron it is possible to attach a natural number so that for each pair of vertices with a common edge, the attached numbers are not relatively prime (i.e. they have common divisors), and with each pair of vertices without a common edge the attached numbers are relatively prime. (Note: there are infinitely many prime numbers.)

1973 Polish MO Finals, 3
Tags: geometry , 3d geometry , combinatorial geometry , polyhedron , parallelepiped
A polyhedron $W$ has the following properties: (i) It possesses a center of symmetry. (ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram. (iii) There is a vertex of $W$ at which exactly three edges meet. Prove that $W$ is a parallelepiped.

2009 Estonia Team Selection Test, 3
Tags: polyhedron , regular polygon , convex , geometry , 3d geometry
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

2017 Polish Junior Math Olympiad Second Round, 5.
Tags: geometry , polyhedron
Does there exist a convex polyhedron in which each internal angle of each of its faces is either a right angle or an obtuse angle, and which has exactly $100$ edges? Justify your answer.

2010 Sharygin Geometry Olympiad, 7
Tags: geometry , polyhedron , cut , plane
Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

1998 Bundeswettbewerb Mathematik, 4
Tags: convex , polyhedron , combinatorial geometry , geometry
Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

Russian TST 2019, P2
Tags: polyhedron , edge , combinatorics
Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

1949 Moscow Mathematical Olympiad, 169
Tags: polyhedron , 3d geometry , geometric construction , geometry
Construct a convex polyhedron of equal “bricks” shown in Figure. [img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]

2004 Estonia Team Selection Test, 6
Tags: pentagon , hexagon , combinatorics , combinatorial geometry , 3d geometry , geometry , polyhedron
Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

2002 Olympic Revenge, 4
Tags: combinatorics , polyhedron , counting
Find all pairs \((m,n)\) of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having \(m\) sides, and the other having \(n\) sides.

1971 IMO Shortlist, 16
Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

1996 Austrian-Polish Competition, 5
Tags: geometry , 3d geometry , sphere , polyhedron , convex
A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

2015 Spain Mathematical Olympiad, 1
Tags: polyhedron , graph coloring , combinatorics
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?

KoMaL A Problems 2018/2019, A. 737
Tags: combinatorics , graph theory , convex , polyhedron , point set , 3-dimensional geometry
$100$ points are given in space such that no four of them lie in the same plane. Consider those convex polyhedra with five vertices that have all vertices from the given set. Prove that the number of such polyhedra is even.

1965 Poland - Second Round, 6
Tags: geometry , 3d geometry , polyhedron
Prove that there is no polyhedron whose every plane section is a triangle.

2009 Oral Moscow Geometry Olympiad, 5
Tags: convex , geometry , polyhedron
Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

Kyiv City MO 1984-93 - geometry, 1986.9.2
Tags: geometry , 3d geometry , polyhedron , rhombus
The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

2008 Oral Moscow Geometry Olympiad, 5
Tags: geometry , polyhedron , 3d geometry
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)? (S. Markelov).

1987 Brazil National Olympiad, 2
Tags: geometry , convex , polyhedron
Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.

1975 Bundeswettbewerb Mathematik, 2
Tags: combinatorial geometry , polyhedron , 3d geometry , edge , geometry
Prove that in each polyhedron there exist two faces with the same number of edges.

2017 Israel Oral Olympiad, 6
Tags: 3d geometry , geometry , polyhedron
What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?

2006 All-Russian Olympiad Regional Round, 10.8
Tags: combinatorics , combinatorial geometry , geometry , 3d geometry , polyhedron
A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

1994 Tuymaada Olympiad, 4
Tags: convex , polyhedron , sphere , geometry
Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

2009 Estonia Team Selection Test, 3
Tags: polyhedron , regular polygon , convex , geometry , 3d geometry
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.