1992

Problem. (tthnew) For $a,b,c$ be reals such as $abc(a+b+c) \geqslant 0.$ Prove that$:$ 

$${\dfrac {1}{abc}}+{\dfrac {573}{500}}\cdot{\dfrac {a+b+c}{ \left( {a}^{2}+{b }^{2}+{c}^{2} \right) ^{2}}}\geqslant {\dfrac {6219}{500 ( a +b+c ) ( ab+bc+ac ) }}$$

 But:

For $a,b,c \geqslant 0.$ The inequality $${\dfrac {1}{abc}}+{\dfrac {k \left( a+b+c \right) }{ \left( {a}^{2}+{b}^ {2}+{c}^{2} \right) ^{2}}}\geqslant {\dfrac {3(3+k)}{ \left( a+b+c \right) \left( ab+bc+ac \right) }} $$ is true when $k \leqslant k_{\max} \approx 1.14654479754656$ is a root of the equation $$720\,{X}^{4}+5176\,{X}^{3}+10043\,{X}^{2}-3328\,X-18432=0.$$

Who have a nice solution for these inequality$?$ 

(Inspired from Math Stack Exchange.)