How do I find :$b a^2+ c b^2+ a c^2$ according to below given conditions?

Question

Let $a,b,c>0$ such that: $a+b+c=5$ and

$ \frac 1a +b+\frac 1b +c+ \frac 1c +a=9/10.$

I have tried to find the value of :$b a^2+ c b^2+ a c^2$ using the above gaven but i didn't succeed however i have added a third equality which is :$(a+b+c)^2=25$ but no result

My question is to find :

$b a^2+ c b^2+ a c^2$ ?

Answer 1

Your two equations imply:

$\frac{9}{10} = \frac{1}{a}+b+\frac{1}{b}+c+\frac{1}{c}+a = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+ (a+b+c) = \frac{1}{a}+\frac{1}{b}+\frac{1}{c} + 5.$

Thus $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{9}{10}-5 = \frac{9}{10}-\frac{50}{10}<0$. Since you also state that $a,b,c>0$, there can be no numbers $a,b,c$ meeting your initial conditions.