<s>[INST] Show that $\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$ Let $n \in N, n=2k+1, and \text{ } \frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$.

Show that $$\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$$

I have tried, but I don't get anything. Can you please give me a hint?
 [/INST] From the original equation, we get:
$$abc=(a+b+c)(ab+bc+ca)$$
which is equivalent to 
$$a^2(b+c)+bc(b+c)+ab(b+c)+ca(b+c)=0$$
$$\implies(b+c)(a+c)(a+b)=0$$
Then, obviously any one of the following must hold:
$$a=-b\\b=-c\\c=-a$$
In any case we can prove the equation $$\frac{1}{a^n+b^n+c^n}=\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}$$
with odd $n$. 
Since if we take $a=-b$
we get 
$$\frac{1}{(-b)^n+b^n+c^n}=\frac{1}{(-b)^n}+\frac{1}{b^n}+\frac{1}{c^n}$$
which is equivalent to
$$\frac{1}{c^n}=\frac{1}{c^n}$$
and this is true.....
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