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Let $x_i,y_i \in \mathbb{R}$. Consider the following set of inequalities. \begin{align} x_1-x_2 &\leqslant y_1 \\ x_2-x_3 &\leqslant y_2 \\ &\vdots \\ x_{n-1}-x_n &\leqslant y_{n-1} \\ x_n-x_1 &\leqslant y_n \\ \end{align} If they are consistent, necessarily $\sum_{i=1}^n y_i \geqslant 0$.

Is that also a sufficient condition of feasibility?

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Hint: consider $x_1=y_1+y_2+\ldots+y_n$, $x_2= y_2+\ldots+y_n$, $x_3= y_3+\ldots+y_n$, ... , $x_n = y_n$.

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