Question

For positive \(a,b,c\) such that \(\frac{a}{2}+b+2c=3\), find the maximum of \(\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.\)

Answer 1

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

Answer 2

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

Answer 3

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.

Answer 4

Not exactly Olympiad rigorous but it works.