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Let a, b, c be positive real numbers such that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1.$$Simplify $$\left( 1 + \frac{a}{b} \right) \left( 1 + \frac{b}{c} \right) \left( 1 + \frac{c}{a} \right) \left( \frac{1}{a + bc} + \frac{1}{b + ac} + \frac{1}{c + ab} \right).$$

Sep 8, 2019

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To be honest, you can let a = 2, b = 3, and c = 6 and solve from there.

$$(1+\frac{2}{3})(1+\frac{3}{6})(1+\frac{6}{2})(\frac{1}{2+3\times 6}+\frac{1}{3+2\times 6}+\frac{1}{6+2\times 3})$$

$$(\frac{5}{3})(\frac{3}{2})(4)(\frac{1}{20}+\frac{1}{15}+\frac{1}{12})$$

$$10\times 0.2 = 2$$.

You are very welcome!

:P

Sep 11, 2019