Where did the square roots go?

Algebra Level 4

\[ \dfrac a{\sqrt{a+bc}} + \dfrac b{\sqrt{b+ac}} + \dfrac c{\sqrt{c+ab}} \geq K \left( \dfrac a{a+1} + \dfrac b{b+1} + \dfrac c{c+1} \right) \]

Let \(a,b\) and \(c\) be positive real numbers whose sum is \(1\) and that they satisfy the inequality above for some constant \(K\).

Let \(x\) denote the value of \(a\) when the equality occurs, find \(K+x\).

Submit your answer to \(2\) decimal places.


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