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

1969 Poland - Second Round, 6
Tags: combinatorics , geometry , 3d geometry , polyhedron , combinatorial geometry
Prove that every polyhedron has at least two faces with the same number of sides.

1965 All Russian Mathematical Olympiad, 070
Tags: polyhedron , geometry , 3d geometry
Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

1967 IMO Shortlist, 5
Tags: geometry , 3d geometry , sphere , polyhedron
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

Kvant 2020, M1387
Tags: combinatorics , polyhedron
An ant crawls clockwise along the contour of each face of a convex polyhedron. It is known that their speeds at any given time are not less than 1 mm/h. Prove that sooner or later two ants will collide. [i]Proposed by A. Klyachko[/i]

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.

2016 Brazil Undergrad MO, 5
Tags: polyhedron
A soccer ball is usually made from a polyhedral fugure model, with two types of faces, hexagons and pentagons, and in every vertex incide three faces - two hexagons and one pentagon. We call a polyhedron [i]soccer-ball[/i] if it is similar to the traditional soccer ball, in the following sense: its faces are \(m\)-gons or \(n\)-gons, \(m \not= n\), and in every vertex incide three faces, two of them being \(m\)-gons and the other one being an \(n\)-gon. [list='i'] [*] Show that \(m\) needs to be even. [*] Find all soccer-ball polyhedra. [/list]

2002 District Olympiad, 3
Tags: geometry , 3d geometry , pyramid , polyhedron , angle
Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.) a) Show that the polyhedron $MDNBP$ is a regular pyramid. b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $

1971 IMO, 2
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.

2012 Tournament of Towns, 1
Tags: 3d geometry , convex polyhedron , polyhedron , geometry
Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

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

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

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.

2014 Sharygin Geometry Olympiad, 22
Tags: geometry , polyhedron , convex , 3-dimensional geometry
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?

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.

1971 IMO Longlists, 49
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.

2019 Romanian Master of Mathematics Shortlist, original P5
Tags: combinatorics , polyhedron , edge
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]

1989 All Soviet Union Mathematical Olympiad, 508
Tags: polyhedron , combinatorial geometry , combinatorics , 3d geometry , even
A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).

1971 IMO Longlists, 50
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.

2007 Sharygin Geometry Olympiad, 5
Tags: geometry , combinatorial geometry , convex , polyhedron , 3-dimensional geometry
Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?

1982 Tournament Of Towns, (028) 2
Tags: polyhedron , 3d geometry , combinatorial geometry , geometry
Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges? $AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$. [img]http://1.bp.blogspot.com/-wTdNfQHG5RU/XVk1Bf4wpqI/AAAAAAAAKhA/8kc6u9KqOgg_p1CXim2LZ1ANFXFiWgnYACK4BGAYYCw/s1600/TOT%2B1982%2BAutum%2BS2.png[/img]

2011 Tournament of Towns, 1
Tags: combinatorial geometry , convex polyhedron , polyhedron , similar triangles , geometry
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

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.

2021 China National Olympiad, 5
Tags: graph theory , combinatorics , polyhedron , even , parity
$P$ is a convex polyhedron such that: [b](1)[/b] every vertex belongs to exactly $3$ faces. [b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices. An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible) Show that the number of times the ant turns left is the same as the number of times the ant turn right.

2007 Sharygin Geometry Olympiad, 6
Tags: geometry , symmetry , axes , polygon , polyhedron
a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

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)