Given any $a,b,c \geq 1$, prove that $a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$ Given any $a,b,c \geq 1$, prove that:
$a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$
I tried using most of the popular inequalities and I didn't end up anywhere. Can anyone guide me through this problem?

$RHS^2 \leq (4a^2+4b^2+4c^2)(a+b+c-3)$, so it suffices to show $a+b+c -3 \leq \frac{1}{4}(a^2+b^2+c^2)$
