USAMO
1999
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.