\[\boxed{\text{256.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[\frac{1}{z - a} + \frac{1}{z - b} = \frac{1}{a} + \frac{1}{b}\]
\[Доказать,\ что\ z = \frac{2}{\frac{1}{a} + \frac{1}{b}}.\]
\[\frac{1}{\frac{2}{\frac{1^{\backslash a}}{a} + \frac{1^{\backslash b}}{b}} - a} + \frac{1}{\frac{2}{\frac{1^{\backslash b}}{a} + \frac{1^{\backslash a}}{b}} - b} =\]
\[= \frac{1}{a} + \frac{1}{b}\]
\[\frac{1}{\frac{2ab}{b + a} - a^{\backslash b + a}} + \frac{1}{\frac{2ab}{b + a} - b^{\backslash b + a}} =\]
\[= \frac{1}{a} + \frac{1}{b}\]
\[\frac{b + a}{2ab - ab - a^{2}} + \frac{b + a}{2ab - ab - b^{2}} =\]
\[= \frac{1}{a} + \frac{1}{b}\]
\[\frac{b + a^{\backslash b}}{ab - a^{2}} + \frac{b + a^{\backslash a}}{ab - b^{2}} = \frac{1}{a} + \frac{1}{b}\]
\[\frac{b + a}{\text{ab}} = \frac{1}{a} + \frac{1}{b}\]
\[\frac{a}{\text{ab}} + \frac{b}{\text{ab}} = \frac{1}{a} + \frac{1}{b}\]
\[\frac{1}{a} + \frac{1}{b} = \frac{1}{a} + \frac{1}{b}\]
\[Что\ и\ требовалось\ доказать.\]