Question

Show for positive number a, b, c $\frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\le a+b+c$

Answer 1

We that AM $\ge$GM

$\frac{\frac{bc}{a}+\frac{ac}{b}}{2}\ge {\left(\frac{ab{c}^2}{ab}\right)}^{\frac{1}{2}}\Rightarrow \frac{bc}{a}+\frac{ac}{b}\ge 2c...\left(i\right)$

Similarly $\frac{ac}{b}+\frac{ab}{c}\ge 2a...\left(ii\right)$

$\frac{ab}{c}+\frac{bc}{a}\ge 2b...\left(iii\right)$

Now add (i), (ii) & (iii)

$2(\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b})\ge 2(a+b+c)$