Đáp án:
`(\sqrt{x - 2009} - 1)/(x - 2009) + (\sqrt{y - 2010} - 1)/(y - 2020) + (\sqrt{z - 2011} - 1)/(z - 2011) = 3/4`
Đặt : `\sqrt{x - 2019} = a`
`\sqrt{y - 2010} = b`
`\sqrt{z - 2011} = c`
phương trinh sẽ thành :
`(a - 1)/a^2 + (b - 1)/b^2 + (c - 1)/c^2 = 3/4`
`<=> 1/a - 1/a^2 + 1/b - 1/b^2 + 1/c - 1/c^2 = 3/4`
`<=> 1/a - 1/a^2 + 1/b - 1/b^2 + 1/c - 1/c^2 - 3/4 = 0`
`<=> (1/a^2 - 2. 1/a . 1/2 + 1/4) + (1/b^2 - 2. 1/b . 1/2 + 1/4) + (1/c^2 - 2 . 1/c . 1/2 + 1/4) = 0`
`<=> (1/a - 1/2)^2 + (1/b - 1/2)^2 + (1/c - 1/2)^2 = 0`
Do ` (1/a - 1/2)^2 ≥ 0`
`(1/b - 1/2)^2 ≥ 0`
`(1/c - 1/2)^2 ≥ 0`
`=> (1/a - 1/2)^2 + (1/b - 1/2)^2 + (1/c - 1/2)^2 ≥ 0`
Dấu "=" xẩy ra
<=> $\left[ \begin{array}{l}\dfrac{1}{a} - \dfrac{1}{2} = 0 \\\dfrac{1}{b} - \dfrac{1}{2} = 0\\\dfrac{1}{c} - \dfrac{1}{2} = 0\end{array} \right.$
<=>$ \left[ \begin{array}{l}\\\sqrt{x - 2009} = 2\\\sqrt{y - 2010} = 2\\\sqrt{z - 2011} = 2\end{array} \right.$
<=>$ \left[ \begin{array}{l}x =2013\\b= 2014\\z = 2015\end{array} \right.$
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