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View attachment 116522
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\[\begin{align}
& x=a;2y=b;3z=c\Rightarrow a+b+c=2 \\
& \Rightarrow S=\sqrt{\frac{ab}{ab+2c}}+\sqrt{\frac{bc}{bc+2a}}+\sqrt{\frac{ac}{ac+2b}} \\
& =\sqrt{\frac{ab}{(a+c)(b+c)}}+\sqrt{\frac{bc}{(a+b)(a+c)}}+\sqrt{\frac{ac}{(a+b)(b+c)}} \\
& \le \frac{1}{4}(\frac{a}{a+c}+\frac{b}{b+c}+\frac{b}{a+b}+\frac{c}{a+c}+\frac{a}{a+b}+\frac{c}{b+c}) \\
& =\frac{3}{4} \\
& ''=''\Leftrightarrow a=b=c=\frac{2}{3}\Leftrightarrow x=\frac{2}{3};y=\frac{1}{3};z=\frac{2}{9} \\
\end{align}\]