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Let $\sigma_{ij}, \ i,j=1,\ldots,n$ ($n \geq 4$) be a sequence of positive real numbers such that $\sigma_{ij}=\sigma_{ji}$. Do you know any sufficient condition on $\sigma_{ij}$ (which is simpler than the system itself) such that the linear system of equations

$$ a_i+a_j=\sigma_{ij},\ i,j=1,\ldots,n, i \neq j $$

admits at least a solution? Thank you.

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    The following "4-cycle condition" is clearly necessary: $\sigma_{ij} + \sigma_{kl} = \sigma_{jk} + \sigma_{li}$ for all distinct ordered 4-tuples $(i, j, k, l)$. I think this is sufficient as well, but I don't have a proof at hand.2012-01-30

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