For positive $a,b,c$ such that $\frac{a}{2}+b+2c=3$, find the maximum of $\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.$
You get the maximum when a = b = c = 6/7. Then min{1/2*ab, ac, 2bc} = 18/49.
