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

2010 Sharygin Geometry Olympiad, 7
Tags: cut , polyhedron , plane , geometry
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?

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]

1985 IMO Longlists, 9
Tags: polyhedron , 3d geometry , euler s theorem , angle , geometry
A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$

2007 Sharygin Geometry Olympiad, 5
Tags: polyhedron , convex , 3-dimensional geometry , combinatorial geometry , 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?

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

1971 IMO Longlists, 50
Tags: geometry , polyhedron , 3d geometry , volume , combinatorial geometry
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.

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]

2009 Estonia Team Selection Test, 3
Tags: regular polygon , polyhedron , 3d geometry , convex , 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.

1967 IMO Longlists, 34
Tags: geometry , polyhedron , 3d geometry , sphere
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.

2019 Romanian Master of Mathematics Shortlist, original P5
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]

2013 Oral Moscow Geometry Olympiad, 3
Tags: polyhedron , area , ratio , 3d geometry , geometry
Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

1988 Tournament Of Towns, (196) 3
Tags: polyhedron , combinatorial geometry , combinatorics
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.)

Kvant 2019, M2573
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]

2004 Estonia Team Selection Test, 6
Tags: polyhedron , pentagon , 3d geometry , hexagon , geometry , combinatorial geometry , combinatorics
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.

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

1999 Tournament Of Towns, 7
Tags: combinatorics , convex polyhedron , polyhedron , combinatorial geometry
Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides. (A Kanel)

2006 All-Russian Olympiad Regional Round, 10.8
Tags: polyhedron , geometry , 3d geometry , combinatorial geometry , combinatorics
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 ?

1990 Tournament Of Towns, (261) 5
Tags: combinatorial geometry , 3d geometry , polyhedron , geometry
Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to (a) no less than $ 5$ faces? (b) exactly $5$ faces? (G. Galperin)

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?

1973 Polish MO Finals, 3
Tags: polyhedron , 3d geometry , parallelepiped , geometry , combinatorial geometry
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.

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

1971 IMO, 2
Tags: combinatorial geometry , geometry , polyhedron , 3d 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.

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

1967 IMO Shortlist, 5
Tags: polyhedron , geometry , 3d geometry , sphere
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.

2022 Vietnam TST, 2
Tags: combinatorics , polyhedron
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.