ILL 1985 Problem 1

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that:

(i) $i$ and $n - i$ always receive the same color, and

(ii) for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$

Prove that all numbers in $N$ must receive the same color.