Olympiad problems

1987 Brazil National Olympiad, 2

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 $F$.

1996 Austrian-Polish Competition, 5

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

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 $P$.

1987 IMO Longlists, 18

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]

2009 Estonia Team Selection Test, 3

Tags: polyhedron , regular polygon , convex , 3d geometry , 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.

2014 Sharygin Geometry Olympiad, 3

Tags: geometry , convex , polyhedron , diagonal

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)

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.

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.

1998 Bundeswettbewerb Mathematik, 4

Tags: convex , polyhedron , combinatorial geometry , 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.

1991 ITAMO, 5

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

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

Russian TST 2019, P2

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]

1985 IMO Shortlist, 2

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 $x/y.$

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.

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.

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.

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 $M$?

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]

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.