Question 54
Let $a$, $b$ and $c$ be non-zero real numbers (with non-zero sum) such that:
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}
$$
Proof that is also verified:
$$
\frac{1}{a^{1999}} + \frac{1}{b^{1999}} + \frac{1}{c^{1999}} = \frac{1}{a^{1999} + b^{1999} + c^{1999}}
$$
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}
$$
Proof that is also verified:
$$
\frac{1}{a^{1999}} + \frac{1}{b^{1999}} + \frac{1}{c^{1999}} = \frac{1}{a^{1999} + b^{1999} + c^{1999}}
$$