## zmudz one year ago For positive $$a,b,c$$ such that $$\frac{a}{2}+b+2c=3$$, find the maximum of $$\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.$$

1. zmudz

$$\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.$$ means the minimum of that set

2. thomas5267

$\frac{1}{2}ab\leq\frac{9}{4},\,a=3\land b=\frac{3}{2}\land c=0\\ ac\leq\frac{9}{4},\,a=3\land b=0\land c=\frac{3}{4}\\ 2bc\leq\frac{9}{4},\,a=0\land b=\frac{3}{2}\land c=\frac{3}{4}$

3. thomas5267

The answer is 1 with $$a=2,b=1,c=\dfrac{1}{2}$$. My idea was that increase in one of the three expression will result in the decrease of another. So $$\dfrac{1}{2}ab=ac=2bc$$ would be nice. Solving this yields the answer. The answer is also verified by Mathematica.

4. thomas5267

Not exactly Olympiad rigorous but it works.