Olympiad problems

2012 Tournament of Towns, 5

In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the first player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?

2013 Tournament of Towns, 4

Tags: Chessboard , Rook , combinatorics

Eight rooks are placed on a $8\times 8$ chessboard, so that no two rooks attack one another. All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.

1999 Tournament Of Towns, 6

Tags: combinatorics , Chessboard , Rook

A rook is allowed to move one cell either horizontally or vertically. After $64$ moves the rook visited all cells of the $8 \times 8$ chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal. (A Shapovalov, R Sadykov)