exercice ramadanne !!!

[img]http://latex.codecogs.com/gif.latex?LHS-RHS=\sum(\frac{a^2}{b^2+c^2}-\sum&space;\frac{a}{b+c})=\sum&space;\frac{a^2(b+c)-a(b^2+c^2)}{(b^2+c^2)(b+c)}=\sum&space;\frac{ab(a-b)+ac(a-c)}{(b+c)(b^2+c^2)}=\sum&space;(\frac{ab(a-b)}{(b+c)(b^2+c^2)}-\frac{ac(c-a)}{(b+c)(b^2+c^2)})=\sum&space;(\frac{ab(a-b)}{(b+c)(b^2+c^2)}-\frac{ba(a-b)}{(c+a)(c^2+a^2)})=\sum&space;ab(a-b)[\frac{(c+a)(a^2+c^2)-(b+c)(c^2+b^2)}{(a+c)(b+c)(c^2+a^2)(b^2+c^2)}]=\sum&space;\frac{ab(a-b)^2(a^2+b^2+c^2+ab+bc+ca)}{(a+c)(b+c)(b^2+c^2)(c^2+a^2)}\geq{0}[/img]