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Part of 2008 Federal Competition For Advanced Students, P1

Problems(1)

nxn system x_1x_2(3a-2x_3) = a^3 ... x_nx_1(3a-2x_2) = a^3

Source: Austrian Federal Competition For Advanced Students 2008, Part 1, p2

8/30/2019
Given a∈R+a \in R^{+} and an integer n>4n > 4 determine all n-tuples (x1,...,xnx_1, ...,x_n) of positive real numbers that satisfy the following system of equations: {x1x2(3a−2x3)=a3x2x3(3a−2x4)=a3...xn−2xn−1(3a−2xn)=a3xn−1xn(3a−2x1)=a3xnx1(3a−2x2)=a3\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases} .
algebrasystem of equations