Let x,y, and z be nonzero complex numbers such that x+y+z=20 and (x−y)2+(x−z)2+(y−z)2=xyz.Find xyzx3+y3+z3.
Solution — click to reveal
We have the factorization x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−xz−yz).Expanding (x−y)2+(x−z)2+(y−z)2=xyz, we get 2x2+2y2+2z2−2xy−2xz−2yz=xyz,so x2+y2+z2−xy−xz−yz=2xyz, and x3+y3+z3−3xyz=20⋅2xyz=10xyz.Then x3+y3+z3=13xyz, so xyzx3+y3+z3=13.