Đáp án:$ M = 2\sqrt{\dfrac{2020}{2021}}$
Giải thích các bước giải:
$ \dfrac{1}{x} + \dfrac{1}{y} - \dfrac{1}{x} = \dfrac{1}{x + y - z} = \dfrac{2020}{2021} (1)$
$ \dfrac{1}{x} + \dfrac{1}{y} - \dfrac{1}{z} = \dfrac{1}{x + y - z} $
$ ⇔ \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{x} + \dfrac{1}{x + y - z} $
$ ⇔ \dfrac{x + y}{xy} = \dfrac{(x + y - z) + z}{z(x + y - z)}$
$ ⇔ xy = z(x + y - z) ⇔ xy = zx + yz - z²$
$ ⇔ z² - yz - zx + xy = 0 ⇔ (z - x)(z - y) = 0$
- Nếu $ z - x = 0 ⇔ z = x $ thay vào $(1): \dfrac{1}{y} = \dfrac{2020}{2021}=> \dfrac{1}{\sqrt{y}} = \sqrt{\dfrac{2020}{2021}} $
$ => M = \dfrac{1}{\sqrt{x}} + \dfrac{1}{\sqrt{y}} - \dfrac{1}{\sqrt{z}} +\dfrac{1}{\sqrt{x + y - z}} =\dfrac{2}{\sqrt{y}} = 2\sqrt{\dfrac{2020}{2021}} $
- Nếu $ z - y = 0 ⇔ z = y $ thay vào $(1): \dfrac{1}{x} = \dfrac{2020}{2021} => \dfrac{1}{\sqrt{x}} = \sqrt{\dfrac{2020}{2021}} $
$ => M = \dfrac{1}{\sqrt{x}} + \dfrac{1}{\sqrt{y}} - \dfrac{1}{\sqrt{z}} +\dfrac{1}{\sqrt{x + y - z}} =\dfrac{2}{\sqrt{x}} = 2\sqrt{\dfrac{2020}{2021}} $
