Prove $\frac{a}{bc}+ \frac{b}{ca}+ \frac{c}{ab} \ge 1$ Let $a,b,c$ be positive real numbers such that $\dfrac{1}{bc}+ \dfrac{1}{ca}+ \dfrac{1}{ab} \ge 1$. Prove that $\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} \ge 1$.

multiply abc
$a+b+c\geq abc $
$a^2+b^2+c^2\geq abc $
$a^2+b^2+c^2-(a+b+c)$
$⇔(a-\frac12)^2+(b-\frac12)^2+(c-\frac12)^2\geq \frac34$
therefore
$a^2+b^2+c^2\geq \frac34+abc$
