Olympiad problems

1965 Poland - Second Round, 6

Prove that there is no polyhedron whose every plane section is a triangle.

2004 Estonia Team Selection Test, 6

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

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.

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.

2012 Tournament of Towns, 1

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

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.

1985 Polish MO Finals, 6

Tags: geometry , 3d 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.

2012 Oral Moscow Geometry Olympiad, 4

Tags: 3d geometry , polyhedron , geometry

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.

2002 Olympic Revenge, 4

Tags: counting , polyhedron , combinatorics

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.

2002 District Olympiad, 3

Tags: 3d geometry , polyhedron , angle , geometry , pyramid

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) $

1971 IMO Longlists, 49

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.

II Soros Olympiad 1995 - 96 (Russia), 11.5

Tags: angle , 3d geometry , polyhedron , geometry

$6$ 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.

2017 Israel Oral Olympiad, 6

Tags: 3d geometry , polyhedron , geometry

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

1987 IMO Shortlist, 4

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

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: angle , 3d geometry , polyhedron , euler s theorem , 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 $x/y.$

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.

2007 Sharygin Geometry Olympiad, 6

Tags: symmetry , geometry , 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?

2019 IFYM, Sozopol, 7

Tags: geometry , polyhedron , combinatorics

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 $m$.

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.

1984 Czech And Slovak Olympiad IIIA, 5

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

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.

1987 IMO Longlists, 18

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

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]

Kyiv City MO 1984-93 - geometry, 1986.9.2

Tags: geometry , 3d geometry , polyhedron , rhombus

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

1989 All Soviet Union Mathematical Olympiad, 508

Tags: combinatorial geometry , combinatorics , polyhedron , 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).

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 $M$?

2018 Iranian Geometry Olympiad, 4

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

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

1998 Bundeswettbewerb Mathematik, 4

Tags: geometry , convex , polyhedron , combinatorial geometry

Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

1975 Bundeswettbewerb Mathematik, 2

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

Prove that in each polyhedron there exist two faces with the same number of edges.