If $a, b, c$ are all non-zero and $a+b+c=0,$ prove that $\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=3$
If $a, b, c$ are all non-zero and $a+b+c=0,$ prove that $\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=3$
We have $a, b, c$ are all non-zero and $a+b+c=0,$ therefore
$a^{3}+b^{3}+c^{3}=3 a b c$
Now, $\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=\frac{a^{3}+b^{3}+c^{3}}{a b c}=\frac{3 a b c}{a b c}=3$
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