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Algebra Level 4

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

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