ĐK: \(x;y\ge2007\)
\(pt\text{ (1)}\Leftrightarrow x+2006+y-2007+2\sqrt{\left(x+2006\right)\left(y-2007\right)}=4013^2\)
\(pt\text{ (2)}\Leftrightarrow x-2007+y+2006+2\sqrt{\left(x-2007\right)\left(y+2006\right)}=4013^2\)
Suy ra: \(\sqrt{\left(x+2006\right)\left(y-2007\right)}=\sqrt{\left(x-2007\right)\left(y+2006\right)}\)
\(\Leftrightarrow xy-2007x+2006y-2006.2007=xy+2006x-2007y-2006.2006\)
\(\Leftrightarrow\left(2006+2007\right)x=\left(2006+2007\right)y\)
\(\Leftrightarrow x=y\)
Thế x = y vào pt thứ nhất, ta được:
\(\sqrt{x+2006}+\sqrt{x-2007}=4013\)
\(\Leftrightarrow\sqrt{x+2006}-2007+\sqrt{x-2007}-2006=0\)
\(\Leftrightarrow\frac{x+2006-2007^2}{\sqrt{x+2006}+2007}+\frac{x-2007-2006^2}{\sqrt{x-2007}+2006}=0\)
\(\Leftrightarrow\left(x-4026043\right)\left(\frac{1}{\sqrt{x+2006}+2007}+\frac{1}{\sqrt{x-2007}+2006}\right)=0\)
\(\Leftrightarrow x=4026043\)\(\Rightarrow y=4026043\)
Kết luận: \(\left(x;y\right)=\left(4026043;4026043\right)\)
