Question 198
Given that $p, q$ and $r$ are positive real numbers, prove that
$$
\frac{1}{q+r} + \frac{1}{r+p} + \frac{1}{p+q} \ge \frac{9}{2(p+q+r)}
$$
Hence prove that if $m$ is a real number greater than $1$, then
$$
\frac{p^m}{q+r} + \frac{q^m}{r+p} + \frac{r^m}{p+q} \ge \frac{(p+q+r)^{m-1}}{2 \cdot 3^{m-2}}
$$
$$
\frac{1}{q+r} + \frac{1}{r+p} + \frac{1}{p+q} \ge \frac{9}{2(p+q+r)}
$$
Hence prove that if $m$ is a real number greater than $1$, then
$$
\frac{p^m}{q+r} + \frac{q^m}{r+p} + \frac{r^m}{p+q} \ge \frac{(p+q+r)^{m-1}}{2 \cdot 3^{m-2}}
$$