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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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Your statement is true for $n=1$ and $n=2$, but probably fails for $n \geq 3$. – N. S. Nov 3 '15 at 13:40
up vote 17 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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