A System of Equations Begging for Generalization

Here is problem #672 from The Pentagon magazine published by Kappa Mu Epsilon. The problem and the solution below are by Jose Luis Diaz-Barrero, Universitat Politecnica de Catalunya, Barcelona, Spain:

Find all positive solutions of the following system of equations:

\( \left\{ \begin{array}{6,2} x_{1} + x_{2} + x_{3} =& x_{4}^{2} \\ x_{2} + x_{3} + x_{4} =& x_{5}^{2} \\ x_{3} + x_{4} + x_{5} =& x_{6}^{2} \\ x_{4} + x_{5} + x_{6} =& x_{1}^{2} \\ x_{5} + x_{6} + x_{1} =& x_{2}^{2} \\ x_{6} + x_{1} + x_{2} =& x_{3}^{2} \end{array} \right. \)

Solution

Find all positive solutions of the following system of equations:

If the problem has a solution, i.e. if there are six positive numbers \(x_{i},\space i=1,\ldots ,6\) then there are among them a smallest and a largest. Assume

\(x_{a}\le x_{i}\le x_{b},\space i=1,\ldots ,6.\)

Then (thinking of the indices as arranged cyclically):

\(3x_{a}\le x_{a-3} + x_{a-2} + x_{a-1} = x_{a}^{2},\)

implying \(x_{a}\ge 3\). Similarly, \(x_{b}\le 3\), from which all six numbers equal \(3\). Now it is immediate that \((3,3,3,3,3,3)\) does indeed solve the system.

This is a curious result that reminds me of equation \(x^{x^x}=3\). The latter could have been modified by adding the number of exponents \(x\) with changing the solution (which, incidentally, was \(x=3\).)

Here, too, the fact that there are \(6\) equations in \(6\) variables is not important. As long as in each equation the left-hand side is the sum of three successive variables, any number of equations \(n\gt 3\) would lead to the same solution. An extra universality is obtained by scaling the variables to be all \(1\)s:

\(x_{i} + x_{i+1} + x_{i+2} = 3x_{i+3}^{2}, \space i=1, ..., n, \space n \gt 3.\)

In this form, a generalization is immediate:

\(x_{i} + x_{i+1} + \ldots + x_{i+k-1} = kx_{i+k}^{2}, \space i=1, ..., n, \space n \gt k.\)

And when push comes to shove, the exponent can also be replaced:

\(x_{i} + x_{i+1} + \ldots + x_{i+k-1} = kx_{i+k}^{\alpha}, \space i=1, ..., n, \space n \gt k, \space\alpha \gt 0.\)

All of these systems succumb to Jose Luis' approach.

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