1999 USAMO 第 1 题

Some checkers placed on an $n\times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

放置在$n\times n$棋盘上的一些棋子满足以下条件：

(a) 每个不包含棋子的方格与包含方格的方格共享一条边；

(b) 给定任意一对包含棋子的方格，存在包含棋子的方格序列，以给定方格开始和结束，使得该序列的每两个连续方格共享一条边。

证明棋盘上至少放置了 $(n^{2}-2)/3$ 个棋子。