Shortlist JBMO, 2007

Solution:

The inequality rewrites as $\sum \frac{2 a^{2}+b c-3 b c}{2 a^{2}+b c} \leq 0$, or $3-3 \sum \frac{b c}{2 a^{2}+b c} \leq 0$ in other words $\sum \frac{b c}{2 a^{2}+b c} \geq 1$.

Using Cauchy-Schwarz inequality we have
$$\sum \frac{b c}{2 a^{2}+b c}=\sum \frac{b^{2} c^{2}}{2 a^{2} b c+b^{2} c^{2}} \geq \frac{\left(\sum b c\right)^{2}}{2 a b c(a+b+c)+\sum b^{2} c^{2}}=1$$
as claimed.