Question 102
Let $k \gt 2$ be a real number.
a) Prove that for all positive real numbers $x$, $y$ and $z$ the following inequality holds:
$$
\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \gt 2\sqrt{\frac{(x+y)(y+z)(z+x)}{xy+yz+zx}}
$$
b) Prove that there exist positive real numbers $x$, $y$ and $z$ such that
$$
\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \lt k \sqrt{\frac{(x+y)(y+z)(z+x)}{xy+yz+zx}}
$$
a) Prove that for all positive real numbers $x$, $y$ and $z$ the following inequality holds:
$$
\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \gt 2\sqrt{\frac{(x+y)(y+z)(z+x)}{xy+yz+zx}}
$$
b) Prove that there exist positive real numbers $x$, $y$ and $z$ such that
$$
\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \lt k \sqrt{\frac{(x+y)(y+z)(z+x)}{xy+yz+zx}}
$$