<s>[INST] MOP 2011 inequality If $a,b,c$ are positive integers prove that 

$\sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} +9(abc)^{1/3} \le 4(a+b+c)$

My attempt:
I tried to split inequality and prove it bit by bit
$9(abc)^{1/3} \le 3(a+b+c)$
It suffices to prove that 
$\sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} \le a+b+c$
After trying out few values I realized it is wrong.So I decided to tackle it all at your once. I tried few well known inequalities and played around but nothing came useful out of it. Any help??
 [/INST] By Jensen $$\sum_{cyc}\sqrt{a^2-ab+b^2}\leq3\sqrt{\frac{\sum\limits_{cyc}(a^2-ab+b^2)}{3}}=\sqrt{3\sum_{cyc}(2a^2-ab)}.$$
Thus, it's enough to prove that:
$$\sqrt{3\sum_{cyc}(2a^2-ab)}+9\sqrt[3]{abc}\leq4(a+b+c).$$
Now, by $uvw$ (see here: https://artofproblemsolving.com/community/c6h278791 ) 
it's enough to prove the last inequality for equality case of two variables.
Can you end it now?
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