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a^2-bc=b^2-ac=c^2-ab=a^3+b^3+c^3; possible values of (a+b+c)

Source: 2024 Girls in Mathematics Tournament, Level B, Problem 1

October 26, 2024
algebra

Problem Statement

The nonzero real numbers a,b,ca,b,c are such that: a2−bc=b2−ac=c2−ab=a3+b3+c3a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3. Compute the possible values of a+b+ca+b+c.
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