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Classic algebra

Source: Finnish Mathematics Competition 2002, Final Round, Problem 2

November 14, 2011
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Problem Statement

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