Olympiad problems

2004 Estonia Team Selection Test, 6

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.

1985 IMO Longlists, 9

Tags: geometry , 3d geometry , polyhedron , euler s theorem , angle

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 $x/y.$

2017 Polish Junior Math Olympiad Second Round, 5.

Tags: geometry , polyhedron

Does there exist a convex polyhedron in which each internal angle of each of its faces is either a right angle or an obtuse angle, and which has exactly $100$ edges? Justify your answer.

2006 All-Russian Olympiad Regional Round, 10.8

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

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 , 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)

2018 Iranian Geometry Olympiad, 4

Tags: 3-dimensional geometry , 3d geometry , polyhedron , centroid , geometry

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

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.

2015 Spain Mathematical Olympiad, 1

Tags: polyhedron , graph coloring , combinatorics

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?

KoMaL A Problems 2018/2019, A. 737

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

$100$ 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.

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 $m$.

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 $2$ ?

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.

2009 Estonia Team Selection Test, 3

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

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 $\frac{V}{S}$.

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)

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]

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.

2010 Sharygin Geometry Olympiad, 7

Tags: geometry , polyhedron , cut , plane

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?

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).

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]

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.)

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.

1973 Polish MO Finals, 3

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

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.

2017 Israel Oral Olympiad, 6

Tags: 3d geometry , geometry , polyhedron

What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?

2002 Olympic Revenge, 4

Tags: combinatorics , polyhedron , counting

Find all pairs $(m,n)$ of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having $m$ sides, and the other having $n$ sides.