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Algebra Level 4

$a,b,$ and $c$ are real numbers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}.$ Does the following equation hold for all odd integers $n?$ $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$