2112

 Problem. (?) For $a,b,c$ be non-negative numbers such as $a \geq 2(b+c).$ Prove:$$5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$$

Solution. (tthnew)

We write the inequality as 

\[\big[2(b+c)-a\big]^2 \big[4\left( {{\mkern 1mu} b + {\mkern 1mu} c} \right){a^2} + 5{\mkern 1mu} \left( {2{\mkern 1mu} c + b} \right)\left( {c + 2{\mkern 1mu} b} \right)a + 3{\mkern 1mu} \left( {b + c} \right)\left( {6{\mkern 1mu} {b^2} + 19{\mkern 1mu} bc + 6{\mkern 1mu} {c^2}} \right)\big]\]

\[+\big[a-2(b+c)\big]\left( {b}^{2}+3bc+{c}^{2} \right)  \left( 36{b}^{2}+77bc+36{ c}^{2} \right)+2bc \left( b+c \right)  \left( b-c \right) ^{2}\geq 0,\]

which is true.

The text you can copy in Github.