Found problems: 67
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]
A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.
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.
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.
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.
Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides.
(A Kanel)
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.
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.
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 $M$?
Construct a convex polyhedron of equal “bricks” shown in Figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]
Prove that every polyhedron has at least two faces with the same number of sides.
$100$ 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.
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.
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.
Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).
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.
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.
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]
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]
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 $m$.
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.
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.
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.
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 $\frac{V}{S}$.
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)