RU-ALLRUSSIAN 1965 Problem 8

The $\mathrm{n}^2$ numbers $\mathbf{x}_{\mathrm{ij}}$ satisfy the $\mathrm{n}^3$ equations: $\mathbf{x}_{\mathrm{ij}} + \mathbf{x}_{\mathrm{jk}} + \mathbf{x}_{\mathrm{ki}} = 0$ . Prove that we can find numbers $\mathbf{a}_1$ , ..., $\mathbf{a}_{\mathrm{n}}$ such that $\mathbf{x}_{\mathrm{ij}} = \mathbf{a}_{\mathrm{i}} - \mathbf{a}_{\mathrm{j}}$ .