<s>[INST] prove that $\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ Assume: $a,b,c >0$ prove that :
$$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
 [/INST] I assume that by $x,y,z$ you mean $a,b,c$.
Without loss of generality, we can assume that $c$ is the smallest of the three i.e. $a, b \geq c >0$. Then
\begin{align}
\dfrac{b+c}{a^2} + \dfrac{c+a}{b^2} + \dfrac{a+b}{c^2} - \dfrac1a - \dfrac1b - \dfrac1c & = \dfrac{b+c-a}{a^2} + \dfrac{c+a-b}{b^2} + \dfrac{a+b-c}{c^2}\\
& = \dfrac{b^3 +b^2c - ab^2 + a^2c + a^3 - a^2b}{a^2b^2} + \dfrac{a+b-c}{c^2}\\
& = \dfrac{c(a^2+b^2)+a^3+b^3-a^2b - ab^2}{a^2b^2} + \dfrac{a+b-c}{c^2}\\
& = \dfrac{c(a^2+b^2)+(a+b)(a^2-ab+b^2)-ab(a+b)}{a^2b^2} + \dfrac{a+b-c}{c^2}\\
& = \dfrac{c(a^2+b^2)+(a+b)(a^2-2ab+b^2)}{a^2b^2} + \dfrac{a+b-c}{c^2}\\
& = \dfrac{c(a^2+b^2)}{a^2b^2}+\dfrac{(a+b)(a-b)^2}{a^2b^2} + \dfrac{a+b-c}{c^2}\\
\end{align}
Note that each term on the right side is non-negative. In fact, the first term (since $a,b,c >0)$ and last term (since $a,b \geq c > 0 \implies a+b > c$) are strictly positive. Hence, $$\dfrac{c(a^2+b^2)+(a+b)(a-b)^2}{a^2b^2} + \dfrac{a+b-c}{c^2} > 0$$ Hence, $$\dfrac{b+c}{a^2} + \dfrac{c+a}{b^2} + \dfrac{a+b}{c^2} - \dfrac1a - \dfrac1b - \dfrac1c > 0$$which gives us more than what we want.
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