Given any $a,b,c \geq 1$, prove that $a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$

Question

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?

Answer 1

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