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Asymmetric polynomials

Source: German TST 2004, exam IV, invented by Eric M�ller, I think

May 18, 2004
algebrapolynomiallinear algebramatrixcomplex numberssystem of equationsalgebra proposed

Problem Statement

Let n be a positive integer. Find all complex numbers x1x_{1}, x2x_{2}, ..., xnx_{n} satisfying the following system of equations: x1+2x2+...+nxn=0x_{1}+2x_{2}+...+nx_{n}=0, x12+2x22+...+nxn2=0x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0, ... x1n+2x2n+...+nxnn=0x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0.
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