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Find all a,b,c in N-Iran 3rd round-Number Theory 2007

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July 28, 2010
symmetryinequalitiesnumber theoryrelatively prime

Problem Statement

Find all a,b,c∈Na,b,c \in \mathbb{N} such that a2b∣a3+b3+c3,b2c∣a3+b3+c3,c2a∣a3+b3+c3.a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.
[PS: The original problem was this: Find all a,b,c∈Na,b,c \in \mathbb{N} such that a2b∣a3+b3+c3,b2c∣a3+b3+c3,c2b∣a3+b3+c3.a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3. But I think the author meant c2a∣a3+b3+c3c^2a|a^3+b^3+c^3, just because of symmetry]
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