[tex]\it S = \dfrac{x_1+x_2}{x_3} + \dfrac{x_2+x_3}{x_1} + \dfrac{x_3+x_1}{x_2}
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S+3 = \dfrac{x_1+x_2}{x_3} +1+ \dfrac{x_2+x_3}{x_1} +1+ \dfrac{x_3+x_1}{x_2}+1 \Rightarrow
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\Rightarrow S+3 = \dfrac{x_1+x_2+x_3}{x_3} + \dfrac{x_1+x_2+x_3}{x_1} + \dfrac{x_1+x_2+x_3}{x_2} [/tex]
[tex]\it S+3 = \dfrac{-5}{x_3} + \dfrac{-5}{x_1} + \dfrac{-5}{x_2} = -5\left(\dfrac{1}{x_3}+\dfrac{1}{x_1}+\dfrac{1}{x_2}\right) \Rightarrow
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\Rightarrow S+3 = -5\cdot \dfrac{x_1x_2 +x_2x_3+x_3x_1}{x_1x_2x_3} \Rightarrow S+3 = -5\cdot\dfrac{0}{4} \Rightarrow
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\Rightarrow S+3 = 0 \Rightarrow S = -3
[/tex]
