Olympiad problems

1977 All Soviet Union Mathematical Olympiad, 241

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.

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 $F$.

1971 IMO Shortlist, 16

Tags: geometry , combinatorial 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.

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]

1994 Tuymaada Olympiad, 4

Tags: geometry , convex , polyhedron , sphere

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 $\frac{V}{S}$.

2014 Sharygin Geometry Olympiad, 3

Tags: convex , polyhedron , diagonal , 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)

2015 Spain Mathematical Olympiad, 1

Tags: combinatorics , polyhedron , graph coloring

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?

1987 Bundeswettbewerb Mathematik, 2

Tags: polyhedron , combinatorics , directed graph , cycle

An arrow is assigned to each edge of a polyhedron such that for each vertex, there is an arrow pointing towards that vertex and an arrow pointing away from that vertex. Prove that there exist at least two faces such that the arrows on their boundaries form a cycle.

2006 IMO Shortlist, 7

Tags: 3d geometry , euler , polyhedron , geometry

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]

1996 Austrian-Polish Competition, 5

Tags: geometry , 3d geometry , polyhedron , convex , sphere

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 $P$.

2009 Estonia Team Selection Test, 3

Tags: geometry , polyhedron , regular polygon , convex , 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.

2009 Oral Moscow Geometry Olympiad, 5

Tags: geometry , convex , polyhedron

Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

1982 Tournament Of Towns, (028) 2

Tags: geometry , combinatorial geometry , polyhedron , 3d 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]

1991 ITAMO, 5

Tags: combinatorial geometry , geometry , polyhedron , 3d geometry

For which values of $n$ does there exist a convex polyhedron with $n$ edges?

KoMaL A Problems 2018/2019, A. 737

Tags: graph theory , convex , polyhedron , point set , 3-dimensional geometry , combinatorics

$100$ 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.

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).

1948 Kurschak Competition, 2

Tags: geometry , 3d geometry , polyhedron , tetrahedron

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.