International Mathematical Olympiad 1985 Problem 2

Let $n$ and $k$ be given relatively prime natural numbers, $k < n$. Each number in the set $M = \{1, 2, \ldots, n-1\}$ is colored either blue or white. It is given that

• (i) for each $i \in M$, both $i$ and $n-i$ have the same color;

• (ii) for each $i \in M, i \neq k$, both $i$ and $|i-k|$ have the same color.

Prove that all numbers in $M$ must have the same color.