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 \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

1969 IMO Longlists, 52

Tags: geometry , polygon , dissection

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.

2008 Hungary-Israel Binational, 3

Tags: geometry , rectangle , combinatorics , dissection

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, 2

Tags: geometry , trapezoid , dissection , cyclic quadrilateral

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

2002 IMO Shortlist, 2

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?

1969 IMO Longlists, 32

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

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