Stronger than Iran TST 1996

     1. For $a,b,c\geqslant 0;ab+bc+ca :\neq 0.$ Prove$:$ 

        $$(ab+bc+ca)\Big(\dfrac{1}{(a+b)^2}+\dfrac{1}{(b+c)^2}+\dfrac{1}{(c+a)^2}\Big) \geqslant \dfrac{9}{4}+\dfrac{ka^2b^2c^2(a-b)^2(a-c)^2(b-c)^2}{(a+b)^4(a+c)^4(b+c)^4}$$

        This inequality is true for all $k \leqslant k_{\max}\approx 3948.42394493992$ is a Root Of: $$3375\,{k}^{5}-18198972\,{k}^{4}+19237295979\,{k}^{3}+13963607218\,{k}^{2}+347614144\,k+1916928=0 \, (1)$$

        See the text of the equation $(1)$ in Github.

     2. (xzlbq) With same condition. Prove$:$

$$(ab+bc+ca)\Big(\dfrac{1}{(a+b)^2}+\dfrac{1}{(b+c)^2}+\dfrac{1}{(c+a)^2}\Big) \geqslant \dfrac{9}{4} \sqrt{\dfrac{(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)}{(ab+bc+ca)(a^2b^2+b^2c^2+c^2a^2)}}$$