Olympiad problems

2018 Iranian Geometry Olympiad, 4

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.) Proposed by Mahdi Etesamifard - Morteza Saghafian

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.

1987 IMO Longlists, 18

Tags: geometric inequality , 3d geometry , polyhedron , geometry

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]

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.

1999 Tournament Of Towns, 7

Tags: convex polyhedron , polyhedron , combinatorics , combinatorial geometry

Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides. (A Kanel)

1985 IMO Longlists, 9

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 $x/y.$

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.

1994 Czech And Slovak Olympiad IIIA, 2

Tags: cube , polyhedron , volume , max , 3d geometry , projection , geometry

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

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]

2012 Oral Moscow Geometry Olympiad, 4

Tags: geometry , 3d geometry , polyhedron

Inside the convex polyhedron, the point $P$ and several lines $\ell_1,\ell_2, ..., \ell_n$ passing through $P$ and not lying in the same plane. To each face of the polyhedron we associate one of the lines $l_1, l_2, ..., l_n$ that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.

2011 Tournament of Towns, 1

Tags: combinatorial geometry , geometry , convex polyhedron , polyhedron , similar triangles

The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

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?

1967 IMO Longlists, 34

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.

2019 IFYM, Sozopol, 7

Tags: combinatorics , geometry , polyhedron

A convex polyhedron has $m$ triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of $m$.

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.

1985 Polish MO Finals, 6

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

There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

1987 Bundeswettbewerb Mathematik, 2

Tags: directed graph , cycle , polyhedron , combinatorics

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.