Problem
ALG-B1-M10-P016 Triple of numbers
#16
★★★☆☆ Level 3 of 5
Find all real triples \((a,b,c)\) such that \(a+b+c=0\), \(a^2+b^2+c^2=8\), \(a^3+b^3+c^3=0\).
Use \(a^3+b^3+c^3=3abc\).
From the zero sum, \(a^3+b^3+c^3=3abc\), so \(abc=0\). Also \(ab+bc+ca=-4\). If one number is zero, say \(c=0\), then \(a+b=0\), \(a^2+b^2=8\), hence \(a=\pm2\), \(b=\mp2\). All solutions are permutations of \((2,-2,0)\).
A mixture of symmetric sums and a complete answer.