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Unhomogenous system of equations

Source: Czech and Slovak Olympiad 2015, National Round, Problem 4

April 1, 2015
algebrasystem of equations

Problem Statement

Find all real triples (a,b,c)(a,b,c), for which a(b2+c)=c(c+ab)a(b^2+c)=c(c+ab) b(c2+a)=a(a+bc)b(c^2+a)=a(a+bc) c(a2+b)=b(b+ca).c(a^2+b)=b(b+ca).
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