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If $x_1,x_2,\dots,x_n \in \mathbb{R}$ can we claim that: $$x_1^3+x_2^3+\dots+x_n^3+x_1+x_2+\dots+x_n=0$$ implies$$x_1+x_2+\dots+x_n=0$$

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up vote 14 down vote accepted

No, we cannot. Here is a counterexample: $n=6$, $x_1=x_2=x_3=x_4=x_5=1$ and $x_6=-2$.

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