jmc

Question

Let a, b, c be distinct real numbers such that ab - c + bc - a + ca - b = 0.Find all possible values of a(b - c)^2 + b(c - a)^2 + c(a - b)^2.Enter all the possible values, separated by commas.

Accepted Answer

Let x = b - c, y = c - a, and z = a - b, so ax + by + cz = 0.Then ( ax + by + cz ) ( 1x + 1y + 1z ) = 0.Expanding, we get ax^2 + by^2 + cz^2 + a + bxy + a + cxz + b + cyz = 0.Note that align* a + bxy + a + cxz + b + cyz &= (a + b)z + (a + c)y + (b + c)xxyz &= (a + b)(a - b) + (a + c)(c - a) + (b + c)(b - c)xyz &= a^2 - b^2 + c^2 - a^2 + b^2 - c^2xyz &= 0, align*so a(b - c)^2 + b(c - a)^2 + c(a - b)^2 = ax^2 + by^2 + cz^2 = 0.