VMO 2017 Problem 4

Given an integer $n>1$ and a $n\times n$ grid $ABCD$ containing $n^2$ unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called symmetry if each unit square has center on diagonal $AC$ is colored by gray and every couple of unit squares which are symmetry by $AC$ should be both colred by black or white. In each gray square, they label a number $0$ , in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called $k$ -balance (with $k\in\mathbb{Z}^+$ ) if it satisfies the following requirements:

i) Each pair of unit squares which are symmetry by $AC$ are labelled with the same integer from the closed interval $[-k,k]$ ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct.

a) For $n=5$ , find the minimum value of $k$ such that there is a $k$ -balance label for the following grid

b) Let $n=2017$ . Find the least value of $k$ such that there is always a $k$ -balance label for a symmetry coloring.