# If a, b, c are all nonzero and a + b + c = 0, prove that

Question:

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

Solution:

$a+b+c=0 \Rightarrow a^{3}+b^{3}+c^{3}=3 a b c$

Thus, we have:

$\left(\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\right)=\frac{a^{3}+b^{3}+c^{3}}{a b c}$

$=\frac{3 a b c}{a b c}$

$=3$