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Let $a, b, c$ be positive real numbers. Show that
$$
\frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}} \geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}
$$
in Mathematics by Diamond (75,914 points) | 15 views

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Best answer
Applying AM-GM on $\frac{a^{2}}{b^{2}}$ and $\frac{b^{2}}{c^{2}}$, we obtain
$$
\frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}} \geq 2 \sqrt{\frac{a^{2} b^{2}}{b^{2} c^{2}}}=2 \frac{a}{c} .
$$
Observe that this last term is one of the terms on the RHS of our inequality, which is a great news for us. Similarly, we have
$$
\begin{array}{l}
\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}} \geq 2 \frac{b}{a} \\
\frac{c^{2}}{a^{2}}+\frac{a^{2}}{b^{2}} \geq 2 \frac{c}{b}
\end{array}
$$
Adding up these 3 inequalities and dividing by 2 , we get that
$$
\frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}} \geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \cdot \square
$$
by Diamond (75,914 points)

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