Olympiad problems

2000 Saint Petersburg Mathematical Olympiad, 11.6

Tags: combinatorics , chess , grids , Coloring , Rooks , tabels , Tables

What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them. [I]Proposed by D. Karpov[/i]

2015 IFYM, Sozopol, 4

Tags: combinatorics , Rooks , Chessboard

In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?

2024 Israel National Olympiad (Gillis), P7

Tags: combinatorics , grid , Rooks , paths , national olympiad

A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in $n$ additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every $n$ and such a choice of mines, starting point, and blue cell) in at most [b](a)[/b] $1.99n+100$ moves? [b](b)[/b] $2n-2\sqrt{3n}+100$ moves? [b]Remark.[/b] In each move, the rook goes in a vertical or horizontal line.

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: combinatorics , combinatorial geometry , square , Chessboard , Chess rook , Rooks

On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?