Fractional Equations A

Algebra Level 4

\[ \begin{cases} x_1 + \dfrac1{x_2} = 4 \\ x_2 + \dfrac1{x_3} =1 \\ x_3 + \dfrac1{x_4} = 4 \\ x_4 + \dfrac1{x_5} = 1 \\ \vdots \\ x_{99} + \dfrac1{x_{100}} = 4 \\ x_{100} + \dfrac1{x_1} = 1 \end{cases} \]

Find all positive solutions of the system of fractional equations above.

Submit your answer as \(x_1 + x_2 + \cdots + x_{100} \).

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