The nonzero real numbers $a,b,c$ are such that: $a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3$. Compute the possible values of $a+b+c$.
Source: 2024 Girls in Mathematics Tournament, Level B, Problem 1
Tags: algebra
The nonzero real numbers $a,b,c$ are such that: $a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3$. Compute the possible values of $a+b+c$.