zmudz
  • zmudz
For positive \(a,b,c\) such that \(\frac{a}{2}+b+2c=3\), find the maximum of \(\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.\)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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zmudz
  • zmudz
\(\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}.\) means the minimum of that set
thomas5267
  • thomas5267
\[ \frac{1}{2}ab\leq\frac{9}{4},\,a=3\land b=\frac{3}{2}\land c=0\\ ac\leq\frac{9}{4},\,a=3\land b=0\land c=\frac{3}{4}\\ 2bc\leq\frac{9}{4},\,a=0\land b=\frac{3}{2}\land c=\frac{3}{4} \]
thomas5267
  • 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.

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thomas5267
  • thomas5267
Not exactly Olympiad rigorous but it works.

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