Found problems: 67
What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?
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.
An ant crawls clockwise along the contour of each face of a convex polyhedron. It is known that their speeds at any given time are not less than 1 mm/h. Prove that sooner or later two ants will collide.
[i]Proposed by A. Klyachko[/i]
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 ?
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.
Prove that there is no polyhedron whose every plane section is a triangle.
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?
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?
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.
Prove that in each polyhedron there exist two faces with the same number of edges.
The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.
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]
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
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)
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.
Is there a polyhedron whose area ratio of any two faces is at least $2$ ?
$P$ 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.