IMO 1985 Problem 2

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

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Proof

It suffices to work on $\mathbb{Z}_k$, where $x$ and $-x$ have the same color, and $x$ and $y$ have the same color if $x+y\equiv n(\bmod k)$. We can then reduce $(n,k)$ to $(k, n\bmod k)$, and the latter are relatively prime. Thus we keep reducing until the smaller number is $1$, where it is clear that all numbers have the same color.