Question 4
(AUS 1), IMO2. Each of the numbers in the set $N=\{1,2,3,\ldots,n-1\}$, where $n\ge3$, 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 $i\in N$, $i\ne j$.
Prove that all numbers in $N$ must receive the same color.
(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 $i\in N$, $i\ne j$.
Prove that all numbers in $N$ must receive the same color.