My Problems #15

Algebra Level 4

\(\sqrt{\dfrac{a^2+b^2}{a+b}}+\sqrt{\dfrac{b^2+c^2}{c+b}}+\sqrt{\dfrac{a^2+c^2}{a+c}} +3 \leq k \left(\sqrt{a+b} + \sqrt{c+b} +\sqrt{a+c} \right) \)

Consider all sets of positive real numbers \(a,b,c\) such that \(ab + bc + ca ≤ 3abc\). What is the smallest value of \(k\) such that the inequality above is fulfilled?

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