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System of equations with n variables

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September 8, 2010
algebrasystem of equationsIMO Shortlist

Problem Statement

Find all solutions of the following system of nn equations in nn variables:  x1∣x1∣−(x1−a)∣x1−a∣=x2∣x2∣,x2∣x2∣−(x2−a)∣x2−a∣=x3∣x3∣, ⋮ xn∣xn∣−(xn−a)∣xn−a∣=x1∣x1∣\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array} where aa is a given number.
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