# 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:

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

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.