Problem

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Tags: PMO, 2009, algebra, Inequality, minimum value



Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.