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

1991 ITAMO, 5
Tags: combinatorial geometry , polyhedron , 3d geometry , geometry
For which values of $n$ does there exist a convex polyhedron with $n$ edges?

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

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?

1998 ITAMO, 2
Tags: combinatorial geometry , polyhedron , 3d geometry , edge , geometry
Prove that in each polyhedron there exist two faces with the same number of edges.

2014 Sharygin Geometry Olympiad, 3
Tags: diagonal , convex , polyhedron , geometry
Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

1984 Czech And Slovak Olympiad IIIA, 5
Tags: combinatorics , combinatorial geometry , geometry , 3d geometry , polyhedron
Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.

Kvant 2019, M2573
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]

2002 District Olympiad, 3
Tags: 3d geometry , pyramid , polyhedron , angle , geometry
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) $

1990 Tournament Of Towns, (261) 5
Tags: combinatorial geometry , geometry , 3d geometry , polyhedron
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)

1971 IMO Longlists, 49
Tags: polyhedron , 3d geometry , combinatorial geometry , volume , 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.

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.

1977 All Soviet Union Mathematical Olympiad, 241
Tags: convex polyhedron , polyhedron , sphere , 3d geometry , geometry
Every vertex of a convex polyhedron belongs to three edges. It is possible to circumscribe a circle around all its faces. Prove that the polyhedron can be inscribed in a sphere.

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.

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 $2$ ?

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]

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

II Soros Olympiad 1995 - 96 (Russia), 11.5
Tags: geometry , 3d geometry , polyhedron , angle
$6$ points are taken on the surface of the sphere, forming three pairs of diametrically opposite points on the sphere. Consider a convex polyhedron with vertices at these points. Prove that if this polyhedron has one right dihedral angle, then it has exactly $6$ right dihedral angles.

1987 IMO Shortlist, 4
Tags: geometry , geometric inequality , 3d geometry , polyhedron
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

1985 IMO Shortlist, 2
Tags: 3d geometry , polyhedron , 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 $x/y.$

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.

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).

2006 IMO Shortlist, 7
Tags: geometry , 3d geometry , euler , polyhedron
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces. [i]Proposed by Kei Irei, Japan[/i]

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).

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]