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Question

Let a, b, and c be nonzero real numbers such that 1a + 1b + 1c = 0. Compute bca^2 + acb^2 + abc^2.

Accepted Answer

We have that x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz).Setting x = 1a, y = 1b, and z = 1c, we get x^3 + y^3 + z^3 - 3xyz = 0,since x + y + z = 0. Then 1a^3 + 1b^3 + 1c^3 = 3abc,so bca^2 + acb^2 + abc^2 = 3.