<s>[INST] strong inquality? Suppose $a,b,c$ are positive real number. Prove that
$$\dfrac{a(b+c)}{(a+b)(a+c)}+\dfrac{b(c+a)}{(b+a)(b+c)}+\dfrac{c(a+b)}{(c+a)(c+b)}\ge \dfrac{(a+b+c)^2}{2(a^2+b^2+c^2)}.$$
I 'd tried to use Cauchy-Schwarz's :
$$LHS=\sum_{\{a,b,c\}}\dfrac{(a(b+c))^2}{a(a+b)(a+c)(b+c)}\ge \dfrac{4(ab+bc+ca)^2}{(a+b+c)(a+b)(b+c)(c+a)}.$$
After that, I used $p,q,r$ tranformation and need to prove
$$\dfrac{4q^2}{p(pq-r)}\ge \dfrac{p^2}{2(p^2-2q)}$$
I wonder if it is a true inequality?
Somebody help me? Or give me some hint?
 [/INST] We write the inequality as
$$2+\frac{8abc}{(a+b)(b+c)(c+a)} \geqslant \frac{(a+b+c)^{2}}{a^2+b^2+c^2}.$$
But
$$(a+b)(b+c)(c+a) \leqslant \frac{8}{9}(a+b+c)(a^2+b^2+c^2).$$
It's remain to prove that
$$2+\frac{9abc}{(a+b+c)(a^2+b^2+c^2)} \geqslant \frac{(a+b+c)^{2}}{a^2+b^2+c^2},$$
equivalent to
$$a^2+b^2+c^2+\frac{9abc}{a+b+c} \geqslant 2(ab+bc+ca).$$
Which is true. The proof is completed.
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