Atrisini vienādojumu:
\[\begin{aligned} & \frac{1}{\sqrt{x-2012}+\sqrt{x-2010}}+\frac{1}{\sqrt{x-2010}+\sqrt{x-2008}}+\ldots+ \\ & +\frac{1}{\sqrt{x-2}+\sqrt{x}}+\frac{1}{\sqrt{x}+\sqrt{x+2}}+\ldots+\frac{1}{\sqrt{x+2010}+\sqrt{x+2012}}=\frac{1}{\sqrt{2}} . \end{aligned}\]
Pārveidojam doto vienādojumu, pareizinot ar saucējam saistīto izteiksmi:
\[\begin{aligned} & \frac{\sqrt{x-2012}-\sqrt{x-2010}}{-2}+\frac{\sqrt{x-2010}-\sqrt{x-2008}}{-2}+\ldots+ \\ & +\frac{\sqrt{x-2}-\sqrt{x}}{-2}+\frac{\sqrt{x}-\sqrt{x+2}}{-2}+\ldots+\frac{\sqrt{x+2010}-\sqrt{x+2012}}{-2}=\frac{1}{\sqrt{2}} \end{aligned}\]
Pareizinot abas puses ar \(2\) un saīsinot vienādos locekļus:\[\begin{aligned} & \sqrt{x+2012}-\sqrt{x-2012}=\sqrt{2} \\ & \sqrt{x+2012}=\sqrt{x-2012}+\sqrt{2} \\ & x+2012=x-2012+2 \sqrt{2(x-2012)}+2 \\ & 4022=2 \sqrt{2(x-2012)} \\ & 2011^{2}=2(x-2012) \\ & x=\frac{2011^{2}}{2}+2012=2024072,5 \end{aligned}\]