Trigonometry in Algebra 1

Algebra Level 5

\(\left\{ \begin{array}{l} 2x + {x^2}y = y \\ 2y + {y^2}z = z \\ 2z + {z^2}x = x \\ \end{array} \right.\)

Let \((x_1,y_1,z_1),(x_2,y_2,z_2),...,(x_n,y_n,z_n)\) be the solution set of the above system of equations.

Find: \(\displaystyle\frac{1}{\pi }\sum\limits_{i = 1}^n {\left( {\arctan \left| {{x_n}} \right| + \arctan \left| {{y_n}} \right| + \arctan \left| {{z_n}} \right|} \right)} \)

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