<s>[INST] Let a,b,c be positive real number, proof. Let a,b,c be positive real number, such that  $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=a+b+c$. Prove that :
$\frac{1}{(2a+b+c)^2}+\frac{1}{(2b+c+a)^2}+\frac{1}{(2c+a+b)^2} \leq \frac{3}{16}$
Can anyone help me how to deal with it?
 [/INST] Since $$\displaystyle \frac{1}{a} + \frac{1}{b} + \frac{1}{c}=a+b+c$$
$$\Rightarrow \frac{ab+bc+ca}{abc \left( a+b+c \right)} = 1 $$
$$\Rightarrow (ab+bc+ca)^2=3abc(a+b+c)$$
$$\Rightarrow \frac{3}{ab+bc+ca} = \frac{ab+bc+ca}{abc \left( a+b+c \right)} = 1$$
Or $$\frac{9}{16 \left( ab+bc+ca \right)} = \frac{3}{16}$$
Need proof $$\dfrac{1}{\left( 2a+b+c \right)^2}+\dfrac{1}{\left( 2b+c+a \right)^2}+ \dfrac{1}{\left( 2c+a+b \right)^2} \le \frac{9}{16 \left( ab+bc+ca \right)} $$
Use Am-Gm have $$\left( 2a+b+c \right)^2 = \left( a+b + a+c \right)^2 \ge 4 \left( a+b \right) \left( a+c \right)$$
So $$\dfrac{1}{\left( 2a+b+c \right)^2}+\dfrac{1}{\left( 2b+c+a \right)^2}+ \dfrac{1}{\left( 2c+a+b \right)^2} \le \frac{a+b+c}{ 2 \left( a+b \right) \left( b+c \right) \left( c+a \right)} \le \frac{9}{16 \left( ab+bc+ca \right)}$$
$$ \Leftrightarrow a^2b+ab^2+b^2c+bc^2+c^2a+ca^2 \ge 6abc$$. This is true follow to inequality AM-GM
</s>