FI-MO 2002 Problem 2

Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also $$\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},$$ provided $n$ is an odd positive integer.