Problem

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 2

Tags: algebra proposed, algebra



Let $ a,b,c$ be non-zero real numbers satisfying \[ \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}.\] Find all integers $ n$ such that \[ \dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}=\dfrac{1}{a^n+b^n+c^n}.\]