#problem collection for students
fickle atlasBOT
edgy spear
mental beacon
Let $ a,b,c>0 $ and $(a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) =11 .$ Prove that
$$ \frac{9-5\sqrt 5}{11} \leq\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{(a+b-c)(b+c-a)(c+a-b)}{abc} \leq \frac{9+5\sqrt 5}{11} $$
Let $ a,b,c>0 $ and $ \frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}= a+b+c .$ Prove that
$$ \frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{(a+b-c)(b+c-a)(c+a-b)}{abc} \geq -\frac{10}{7} $$