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System of equations, n variables, n^3-n^2

Source: Austrian-Polish 2005, Problem 7

July 5, 2015
algebrasystem of equationsnumber theory

Problem Statement

For each natural number n≥2n\geq 2, solve the following system of equations in the integers x1,x2,...,xnx_1, x_2, ..., x_n: (n2−n)xi+(∏j≠ixj)S=n3−n2,∀1≤i≤n(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n where S=x12+x22+⋯+xn2.S=x_1^2+x_2^2+\dots+x_n^2.
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