Question 150
Given positive real numbers $a$, $b$, $c$ such that $abc = 1$, prove that for all the integers $k \ge 2$,
$$
\frac{a^k}{a+b} + \frac{b^k}{b+c} + \frac{c^k}{c+a} \ge \frac{3}{2}.
$$
$$
\frac{a^k}{a+b} + \frac{b^k}{b+c} + \frac{c^k}{c+a} \ge \frac{3}{2}.
$$