parmenides51 wrote: Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$. Show that $$\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$$ If $a, b, c, d>0, a+b+c+d=4$ prove that $$\sqrt{a^a+b^b+c^c}+\sqrt{b^b+c^c+d^d}+\sqrt{c^c+d^d+a^a}+\sqrt{d^d+a^a+b^b}\geq4\sqrt{3}$$