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How do you prove that for all positive $a,b,c$ this formula holds true:

\begin{equation*} \frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}? \end{equation*}

Any help will be invaluable.

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$$a+b+c=\sqrt{(a+b+c)^2}=\sqrt{a+b+c}\ \sqrt{a+b+c}\iff \dfrac{a+b+c}{\sqrt{a+b+c}}=\sqrt{a+b+c}$$





we have

$$\dfrac{a}{\sqrt{a+b}}+\dfrac{b}{\sqrt{b+c}}+\dfrac{c}{\sqrt{c+a}}>\dfrac{a+b+c}{% \sqrt{a+b+c}}=\sqrt{a+b+c}.$$

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