Question 78
Suppose $a$, $b$, $c$ and $d$ are positive real numbers such that $\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1} + \frac{1}{d+1} = 2$. Prove that
$$
\sqrt{\frac{a^2+1}{2}} + \sqrt{\frac{b^2+1}{2}} + \sqrt{\frac{c^2+1}{2}} + \sqrt{\frac{d^2+1}{2}} \geq 3(\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}) - 8.
$$
$$
\sqrt{\frac{a^2+1}{2}} + \sqrt{\frac{b^2+1}{2}} + \sqrt{\frac{c^2+1}{2}} + \sqrt{\frac{d^2+1}{2}} \geq 3(\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}) - 8.
$$