Question

If a, b, c are all nonzero and a + b + c = 0, prove that $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}=3$.

Answer 1

$a+b+c=0\Rightarrow a^3+b^3+c^3=3abc$

Thus, we have:
$\left(\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}\right)=\frac{a^3+b^3+c^3}{abc}$
$$\begin{gathered} =\frac{3abc}{abc} \\ =3 \end{gathered}$$