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x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 system

Source: Austrian - Polish 1996 APMC

May 3, 2020
system of equationsalgebra

Problem Statement

Given natural numbers n>k>1n > k > 1, find all real solutions x1,...,xnx_1,..., x_n of the system xi3(xi2+xi+12+...+xi+k−12)=xi−12x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2 for 1 ≤i≤n\le i \le n. Here xn+i=xix_{n+i} = x_i for alli i.
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