Question 74
Let $x$, $y$ and $z$ be positive real numbers such that $x y + y z + z x = 3$.
Prove that
$$
\frac{x+3}{y+z} + \frac{y+3}{z+x} + \frac{z+3}{x+y} + 3 \geqslant 27 \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
$$
Prove that
$$
\frac{x+3}{y+z} + \frac{y+3}{z+x} + \frac{z+3}{x+y} + 3 \geqslant 27 \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^{2}}{(x+y+z)^{3}}
$$