<s>[INST] $min\{ \sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{a+b}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}}\}$ There is an Olympiad problem: $$a,b,c \in \mathbb{R}^+,M=\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{a+b}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}},$$ find the minimum value of $M$.
I think $M$ is minimum when $a=b=c$, but I can't prove it.
Some idea to prove? Or disprove?
Maybe AM-GM work on this problem?
 [/INST] Hint: As the expression is homogeneous, we may set $a+b+c=1$, say. Then
$$M = \sum_{cyc}\left( \sqrt[4]{\frac{a}{1-a}}+\sqrt{\frac{1-a}a}\right)$$
Now you are summing a convex function, so use Jensen's inequality to find the minimum...
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