Olympiad problems

1985 IMO Longlists, 26

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

1981 IMO Shortlist, 10

Tags: dissection , number theory , square , induction

Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

1972 IMO, 2

Tags: dissection , trapezoid , geometry , cyclic quadrilateral

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

2010 Germany Team Selection Test, 2

Tags: dissection , chessboard , perimeter , rectangle , geometry , combinatorics

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2009 IMO Shortlist, 4

Tags: perimeter , dissection , chessboard , combinatorics , rectangle , geometry

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2003 Germany Team Selection Test, 3

Tags: tiling , dissection , combinatorics

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

2008 Germany Team Selection Test, 3

Tags: dissection , rectangle , geometry , combinatorics

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

2008 Hungary-Israel Binational, 3

Tags: rectangle , dissection , geometry , combinatorics

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

1972 IMO Shortlist, 10

Tags: cyclic quadrilateral , trapezoid , dissection , geometry

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

1969 IMO Longlists, 32

Tags: combinatorics , 3d geometry , dissection , sphere , geometry

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

1974 IMO Shortlist, 11

Tags: dissection , geometry , chessboard , rectangle , combinatorics

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]

1969 IMO Shortlist, 52

Tags: polygon , dissection , geometry

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

2007 IMO Shortlist, 2

Tags: dissection , geometry , combinatorics , rectangle

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

1985 IMO Shortlist, 15

Tags: geometry , square , dissection

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

1972 IMO Longlists, 27

Tags: trapezoid , dissection , geometry , cyclic quadrilateral

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

2002 IMO Shortlist, 2

Tags: tiling , dissection , combinatorics

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

2010 Germany Team Selection Test, 2

Tags: rectangle , perimeter , dissection , chessboard , geometry , combinatorics

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

1969 IMO Longlists, 52

Tags: polygon , dissection , geometry

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

1974 IMO Longlists, 1

Tags: combinatorics , rectangle , chessboard , dissection , geometry

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]

1974 IMO, 4

Tags: combinatorics , rectangle , chessboard , dissection

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\ldots <a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\ldots ,a_p$.

2003 Germany Team Selection Test, 3

Tags: combinatorics , tiling , dissection

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

1979 IMO Longlists, 1

Tags: rhombus , polygon , dissection , geometry

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

2003 Kazakhstan National Olympiad, 7

Tags: tiling , dissection , combinatorics

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

1990 IMO Longlists, 20

Tags: geometry , partition , dissection

Could the three-dimensional space be expressed as the union of disjoint circumferences?

1967 IMO Shortlist, 4

Tags: geometry , perimeter , square , dissection , triangulation

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.