IMO Shortlist 1985 problem 4

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Dodao/la: arhiva
April 2, 2012
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.
Source: Međunarodna matematička olimpijada, shortlist 1985