Olympiad problems

1969 IMO Longlists, 52

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: geometry , rectangle , combinatorics , chessboard , dissection

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.

1985 IMO Shortlist, 15

Tags: geometry , dissection , square

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

1974 IMO Shortlist, 11

Tags: geometry , rectangle , combinatorics , chessboard , dissection

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]

1972 IMO Shortlist, 10

Tags: geometry , trapezoid , dissection , cyclic quadrilateral

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