<s>[INST] What is the minimum possible value of $(a + b + c)$? $a$, $b$ and $c$ are real positive numbers satisfying


*

*$\dfrac{1}{3}  \le  ab + bc + ca \le 1$ and

*$abc \ge \dfrac{1}{27}$


What is the minimum possible value of $(a + b + c)$?
 [/INST] By the AM-GM inequality, $\sqrt[3]{abc}\leq \frac{a+b+c}{3}$.
Since $abc \geq \frac{1}{27}$, this implies that $\sqrt[3]{abc}\geq \frac{1}{3}$.
So $\frac{1}{3} \leq \frac{a+b+c}{3}$ and hence $1 \leq a+b+c$.
This minimum value is achieved in the symmetric case where $a=b=c=\frac{1}{3}$.
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