|Time allowed 4 hours 30 min.|
|Each problem is worth 7 points|
Determine all finite sets of at least three points in the plane which satisfy the following condition:
Let be a fixed integer, with .
Consider an square board, where is a fixed even positive integer. The board is divided into unit squares. We say that two different squares on the board are adjacent if they have a common side.
unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.
Determine the smallest possible value of .