Question 85
A finite set of lines, any three non-competing, partition the plan into a number of regions. Two regions are called "neighbors" if their borders share more than a non-degenerate segment. An integer must be written in each region so that: (i) the product of the numbers written in any two neighboring regions is strictly less than their sum; (ii) for each of the given lines, the sum of the numbers written in the regions of the same side of it is equal to $0$. . Prove that this is possible if and only if not all lines are parallel.