Found problems: 20
What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?
Cut the non-equilateral triangle into four similar triangles, among which not all are the same.
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:
a) five?
b) four?
Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1, V_2,\ldots, V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\le i\le n$ such that $V_i$ separates the two vertices.
A partition of the vertices of $G$ into two disjoint non-empty sets is called a [i]minimal cut[/i] if the number of edges crossing the partition is minimal.
[i]Proposed by András Imolay, Budapest[/i]
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons?
by N.Beluhov
Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?
(D. Shnol)
Find all rectangles that can be cut into $13$ equal squares.
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?
At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?
Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed.
(Nairi Sedrakyan)
Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.
Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.
Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size.
(AV Savkin)
In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.
Is it possible to cut a regular triangular prism into two equal pyramids?
In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?
* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.
A triangle can be cut into three similar triangles.
Prove that it can be cut into any number of triangles similar to each other.
Quadrilateral can be cut into two isosceles triangles in two different ways.
a) Can this quadrilateral be nonconvex?
b) If given quadrilateral is convex, is it necessarily a rhomb?