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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

August 30, 2019

algebrasystem of equations

Problem Statement

Given

a \in R^{+}

and an integer

n > 4

determine all n-tuples (

x_1, ...,x_n

) of positive real numbers that satisfy the following system of equations:

\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}