TSTST 2011 Problem 6

Let $a, b, c$ be real numbers in the interval $[0,1]$ with $a+b, b+c, c+a \geq 1$. Prove that $$1 \leq(1-a)^{2}+(1-b)^{2}+(1-c)^{2}+\frac{2 \sqrt{2} a b c}{\sqrt{a^{2}+b^{2}+c^{2}}}$$