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2019 JBMO TST- North Macedonia, problem 4

Source: 2019 JBMO TST- North Macedonia

May 27, 2019
JMMOMacedonia2019Junioralgebra

Problem Statement

Let the real numbers aa, bb, and cc satisfy the equations (a+b)(b+c)(c+a)=abc(a+b)(b+c)(c+a)=abc and (a9+b9)(b9+c9)(c9+a9)=(abc)9(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9. Prove that at least one of aa, bb, and cc equals 00.
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