KöMaL problem A, 2017

Number theory Difficulty 7.0 National Olympiad, round 2 Find the answer Hungary

A positive integer $n$ satisfies the following: it is possible to select some integers such that if we randomly choose two different integers from this list, say, $i$ and $j$ , then $i+j$ $\mathrm{mod\ } n$ is equal to one of the numbers $0,1,\dots,n-1$ with equal probability. Find all numbers $n$ with this property.

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