\(\frac{1}{2x}+\frac{1}{2y}+\frac{1}{xy}=\frac{1}{2}\)
\(\Rightarrow\frac{y}{2xy}+\frac{x}{2xy}+\frac{2}{2xy}=\frac{1}{2}\)
\(\Rightarrow\frac{y+x+2}{2xy}=\frac{1}{2}\)
\(\Rightarrow2\left(y+x+2\right)=2xy\)
\(\Rightarrow y+x+2=xy\)
\(\Rightarrow xy-y-x=2\)
\(\Rightarrow y\left(x-1\right)-\left(x-1\right)=3\)
\(\Rightarrow\left(x-1\right)\left(y-1\right)=3\)
Vì x, y nguyên dương nên \(\left[\begin{matrix}\left\{\begin{matrix}x-1=1\\y-1=3\end{matrix}\right.\\\left\{\begin{matrix}x-1=3\\y-1=1\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow\left[\begin{matrix}\left\{\begin{matrix}x=2\\y=4\end{matrix}\right.\\\left\{\begin{matrix}x=4\\y=2\end{matrix}\right.\end{matrix}\right.\)
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