Question 166
Given a polynomial $f(x) = x^{4} + a x^{3} + b x^{2} + c x$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_{1} + x_{2} = x_{3} + x_{4}$, then the same equality holds for the roots of the second equation.