Neighbors in the garden - Iran NMO 2009 - Problem 2
Source:
September 20, 2010
combinatorics proposedcombinatorics
Problem Statement
In some of the squares of a square garden we've grown apple, pomegranate and peach trees (At most one tree in each square). We call a square a room and call two rooms neighbor if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which there’s no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room.
Prove that the number of empty rooms is not greater than