IMO Shortlist 2000 problem C4
Dodao/la:
arhiva2. travnja 2012. Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be positive integers such that
![\frac{1}{2} n < k \leq \frac{2}{3} n.](/media/m/4/9/6/496eba13cd157faf77fc33997bbc2dfc.png)
Find the least number
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
for which it is possible to place
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
pawns on
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
squares of an
![n \times n](/media/m/9/d/8/9d8eac5b3234425afb9f970edbfe93ef.png)
chessboard so that no column or row contains a block of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
adjacent unoccupied squares.
%V0
Let $n$ and $k$ be positive integers such that $\frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $m$ for which it is possible to place $m$ pawns on $m$ squares of an $n \times n$ chessboard so that no column or row contains a block of $k$ adjacent unoccupied squares.
Izvor: Međunarodna matematička olimpijada, shortlist 2000