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.
Proposed by D. Karpov
We fill all squares with rooks and remove those that do not satisfy the condition. Note that a rook can attack exactly $2,3$ or $4$ rooks on the board. We remove the ones that attack $3$, which are located on the edges (not on the corners). This removes $6 \cdot 4=24$ rooks. Iterating this result again until we get to the center leaves $16$ rooks remaining, which is the maximum. $\blacksquare$