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Let $n$ be a positive integer, $\lambda_{i}$ real numbers and $a_{i}$ for $1\leq i \leq n$ pairwise distinct complex numbers. Help me to prove that if $\forall z \in \mathbb{C}$ we have $ \sum(\lambda_{i}|z-a_{i}|)=0 $, then $\lambda_{i}=0$ for $ 1\leq i \leq n$.

Regards

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    You are free to pick your $z$ as you like. Do so: pick about $n+1$ of them, and see what the resulting system of linear equations says about the $\lambda_i$.2011-04-25
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    @Gunnar : I tried in vain :(2011-04-25
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    @user10049 : It is a tricky one. I'll post a more detailed answer in a couple of minutes. (For now, a more detailed hint. If you want an answer, I'll give it, but you'll profit more from working through the hint.)2011-04-25

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