Question 44
Let $a$, $b$ and $c$ be positive real numbers. Prove that
$$
\frac{3}{2} \lt \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} \lt 9. \quad (\text{Japan 2009})
$$
$$
\frac{3}{2} \lt \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} \lt 9. \quad (\text{Japan 2009})
$$