Atrisināt vienādojumu:
\[\begin{aligned} & \frac{1}{\sqrt{x-2011}+\sqrt{x-2009}}+\frac{1}{\sqrt{x-2009}+\sqrt{x-2007}}+\ldots \\ & +\frac{1}{\sqrt{x-1}+\sqrt{x+1}}+\ldots+\frac{1}{\sqrt{x+2009}+\sqrt{x+2011}}=\frac{1}{\sqrt{2}} \end{aligned}\]
Pārveidojam doto vienādojumu, pareizinot ar saucējam saistīto izteiksmi:
\[\begin{aligned} & \frac{\sqrt{x-2011}-\sqrt{x-2009}}{-2}+\frac{\sqrt{x-2009}-\sqrt{x-2007}}{-2}+\ldots \\ & +\frac{\sqrt{x-1}-\sqrt{x+1}}{-2}+\ldots+\frac{\sqrt{x+209}-\sqrt{x+2011}}{-2}=\frac{1}{\sqrt{2}} \end{aligned}\]
Pareizinot abas puses ar \(2\) un saīsinot vienādos locekļus:\[\begin{aligned} & \sqrt{x+2011}-\sqrt{x-2011}=\sqrt{2} \\ & \sqrt{x+2011}=\sqrt{x-2011}+\sqrt{2} \\ & x+2011=x-2011+2 \sqrt{2(x-2011)}+2 \\ & 4020=2 \sqrt{2(x-2011)} \\ & 2010^{2}=2(x-2011) \\ & x=\frac{2010^{2}}{2}+2011=2010 \cdot 1005+2011=2022061 \end{aligned}\]