<s>[INST] If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$ The following problem is from CHKMO 2018 Problem 1:

If $ab+bc+ca\ge1$, prove that $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$$

I tried to use Cauchy–Schwarz inequality, by try multiplying different things, such as $1^2+1^2+1^2$, $(abc)^2+(abc)^2+(abc)^2$. But I still can’t solve it. Can someone help me?
 [/INST] Just as what was done in Post 1, we can substitute;
$$a=\frac{1}{x}$$
$$b=\frac{1}{y}$$
$$c=\frac{1}{z}.$$
Then, we have that;
\begin{align*}
& \text{ } \text{ } \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } ab+bc+ca \geq 1\\
&\implies \frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\geq 1\\
&\implies \frac{xyz}{xy}+\frac{xyz}{yz}+\frac{xyz}{zx}\geq xyz\\
&\implies x+y+z\geq xyz.\hspace{7.7cm} \rightarrow 1
\end{align*}
We can now proceed as in this link Linked Question
We know by the QM-AM inequality that;
\begin{align*}
\sqrt{\frac{x^2+y^2+z^2}{3}}
&\geq \left(\frac{x+y+z}{3}\right)^2\\
&\geq \left(\frac{xyz}{3}\right)^2.\hspace{2cm} \left(\text{Note that }x+y+z\geq xyz\right)\\
\end{align*}
Squaring both sides and multiplying by 3 gives;
$$x^2+y^2+z^2\geq \frac{x^2y^2z^2}{3}$$
\begin{align*}
\implies\left(x^2+y^2+z^2\right)^\frac{1}{4}
&\geq \left(\frac{x^2y^2z^2}{3}\right)^\frac{1}{4}\\
&\geq \frac{\sqrt{xyz}}{3^\frac{1}{4}}.\hspace{6cm} \rightarrow 2\\
\end{align*}
By AM-GM, we know that;
$$\frac{x^2+y^2+z^2}{3}\geq \sqrt[3]{x^2y^2z^2}$$
$$\implies x^2+y^2+z^2\geq 3\sqrt[3]{x^2y^2z^2}$$
$$\implies \left(x^2+y^2+z^2\right)^3\geq 3^3\left(x^2y^2z^2\right)$$
\begin{align*}
&\implies \left(x^2+y^2+z^2\right)^\frac{3}{4}\geq 3^\frac{3}{4}\sqrt[4]{x^2y^2z^2}\\
&\implies \left(x^2+y^2+z^2\right)^\frac{3}{4}\geq 3^\frac{3}{4}\sqrt{xyz}.\hspace{5.05cm} \rightarrow 3
\end{align*}
We can now multiply Inequality 2 and 3 to have;
\begin{align*}
x^2+y^2+z^2
&=\left(x^2+y^2+z^2\right)^\frac{1}{4}\times \left(x^2+y^2+z^2\right)^\frac{3}{4}\\
&\geq \frac{\sqrt{xyz}}{3^\frac{1}{4}}\times 3^\frac{3}{4}\sqrt{xyz}\\
&\geq xyz\sqrt{3}.
\end{align*}
Substituting back the original variables yields the desired result;
$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\geq\sqrt{3}\frac{1}{abc}$$
$$\implies\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\geq\frac{\sqrt{3}}{abc}.$$
</s>