<s>[INST] If $a,b,c > 0$ and $a+b+c = 1$, then $(\tfrac{1}{a} −1)(\tfrac{1}{b} −1)(\tfrac{1}{c} −1) \geq 8$ 
If $a, b, c$ are positive real numbers and $a+b+c = 1$, prove that
  $$\left(\frac{1}{a} −1\right)\left(\frac{1}{b} −1\right)\left(\frac{1}{c} −1\right) \geq 8.$$

Thank you.
 [/INST] Hint: $\dfrac1a-1=\dfrac{b+c}{a}\ge 2\dfrac{\sqrt{bc}}{a}$.
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