Olympiad problems

2005 Sharygin Geometry Olympiad, 10

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.

2008 Oral Moscow Geometry Olympiad, 1

Tags: cut , geometry , similar triangles

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)

2010 Sharygin Geometry Olympiad, 7

Tags: cut , polyhedron , plane , geometry

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?

2011 Kyiv Mathematical Festival, 3

Tags: cut , quadrilateral , geometry

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?

2006 Sharygin Geometry Olympiad, 8.2

Tags: polygon , cut , geometry , combinatorial geometry , minimum

What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

2016 Sharygin Geometry Olympiad, 4

Tags: cut , polygon , regular polygon , geometry

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

2007 Sharygin Geometry Olympiad, 5

Tags: area , cut , polygon , geometry

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?

2014 Oral Moscow Geometry Olympiad, 2

Tags: cut , geometry , pyramid , 3d geometry , prism

Is it possible to cut a regular triangular prism into two equal pyramids?

2019 Tournament Of Towns, 3

Tags: bicentric quadrilateral , tiling , cut , combinatorial geometry , combinatorics

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

1954 Moscow Mathematical Olympiad, 264

Tags: cut , cube , unfolding , square

* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.

KoMaL A Problems 2024/2025, A. 888

Tags: cut , graph theory , combinatorics

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]

2005 Sharygin Geometry Olympiad, 9.2

Tags: geometry , cut , isosceles triangle , isosceles

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

2006 Sharygin Geometry Olympiad, 10

Tags: diagonal , regular polygon , cut , isosceles , geometry , combinatorial geometry

At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

2017 Yasinsky Geometry Olympiad, 4

Tags: trapezoid , geometry , trapezium , cut

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.

1991 All Soviet Union Mathematical Olympiad, 548

Tags: geometry , cut , transformation , square

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?