ĐKXĐ: \(2019\le x\le2020\)

\(VT=\sqrt{x-2019}+\sqrt{2021-x}\le\sqrt{2\left(x-2019+2021-x\right)}=2\)

\(VP=\left(x-2020\right)^2+2\ge2\)

Đẳng thức xảy ra khi và chỉ khi:

\(\left\{{}\begin{matrix}x-2019=2021-x\\x-2020=0\end{matrix}\right.\) \(\Leftrightarrow x=2020\)

a, \(x^2+\sqrt{x+2021}=2021\) ĐK \(x\ge-2021\)

<=> \(x^2-2021=-\sqrt{x+2021}\)

Đặt \(\sqrt{x+2021}=a\left(a\ge0\right)\)

=> \(\left\{{}\begin{matrix}x^2-2021=-a\\a^2-2021=x\end{matrix}\right.\)

=> \(\left(x-a\right)\left(x+a\right)+a+x=0\)

<=> \(\left[{}\begin{matrix}x+a=0\\x-a+1=0\end{matrix}\right.\)

+ \(x+a=0\)

=> \(\sqrt{x+2021}=-x\)

=> \(\left\{{}\begin{matrix}x\le0\\x^2-x-2021=0\end{matrix}\right.\)=> \(x=\frac{1-7\sqrt{165}}{2}\)

+ \(x-a+1=0\)

=> \(x+1=\sqrt{x+2021}\)

=> \(\left\{{}\begin{matrix}x\ge-1\\x^2+x-2020\end{matrix}\right.\)=> \(x=\frac{-1+\sqrt{8081}}{2}\)

Vậy \(S=\left\{\frac{-1+\sqrt{8081}}{2};\frac{1-7\sqrt{165}}{2}\right\}\)

Ta có: VT = \(\dfrac{x+1}{2021}\)+1 - (\(\dfrac{x+2}{2020}\)+1) = \(\dfrac{x+3}{2019}\)+1=VP
=>\(\dfrac{x+2022}{2021}+\dfrac{x+2022}{2020}-\dfrac{x+2022}{2019}=0\)
=>\(\left(x+2022\right)\left(\dfrac{1}{2021}+\dfrac{1}{2020}-\dfrac{1}{2019}\right)=0\)
=>x +2022 = 0=> x =-2022

:v e xin kiếu

Em tham khảo nhé

https://hoc24.vn/cau-hoi/cho-xsqrtx22021ysqrty220212021tinh-axy.332667728355

ĐKXĐ : \(\left\{{}\begin{matrix}x>2019\\y>2020\\z>2021\end{matrix}\right.\)

Đặt \(\sqrt{x-2019}=a,......\)

Ta được PT : \(\dfrac{1-a}{a^2}+\dfrac{1-b}{b^2}+\dfrac{1-c}{c^2}+\dfrac{3}{4}=0\)

\(\Leftrightarrow\dfrac{1}{a^2}-\dfrac{1}{a}+\dfrac{1}{4}+\dfrac{1}{b^2}-\dfrac{1}{b}+\dfrac{1}{4}+\dfrac{1}{c^2}-\dfrac{1}{c}+\dfrac{1}{4}=0\)

\(\Leftrightarrow\left(\dfrac{1}{a}-\dfrac{1}{2}\right)^2+\left(\dfrac{1}{b}-\dfrac{1}{2}\right)^2+\left(\dfrac{1}{c}-\dfrac{1}{2}\right)^2=0\)

- Thấy : \(\left(\dfrac{1}{a}-\dfrac{1}{2}\right)^2\ge0,......\)

\(\Rightarrow\left(\dfrac{1}{a}-\dfrac{1}{2}\right)^2+\left(\dfrac{1}{b}-\dfrac{1}{2}\right)^2+\left(\dfrac{1}{c}-\dfrac{1}{2}\right)^2\ge0\)

- Dấu " = " xảy ra <=> \(\left\{{}\begin{matrix}\dfrac{1}{a}=\dfrac{1}{2}\\\dfrac{1}{b}=\dfrac{1}{2}\\\dfrac{1}{c}=\dfrac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=2\\b=2\\c=2\end{matrix}\right.\)

- Thay lại a. b. c ta được : \(\left\{{}\begin{matrix}\sqrt{x-2019}=2\\\sqrt{y-2020}=2\\\sqrt{z-2021}=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-2019=4\\y-2020=4\\z-2021=4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=2023\\y=2024\\z=2025\end{matrix}\right.\) ( TM )

Vậy ...

Nhân 2 vế của giả thiết với \(\sqrt{x^2+2021}-x>0\):

\(\left(\sqrt{x^2+2021}-x\right)\left(x+\sqrt{x^2+2021}\right)\left(y+\sqrt{y^2+2021}\right)=2021\left(\sqrt{x^2+2021}-x\right)\)

\(\Leftrightarrow2021\left(y+\sqrt{y^2+2021}\right)=2021\left(\sqrt{x^2+2021}-x\right)\)

\(\Leftrightarrow y+\sqrt{y^2+2021}=\sqrt{x^2+2021}-x\) (1)

Tương tự, nhân 2 vế giả thiết với \(\sqrt{y^2+2021}-y\) và rút gọn ta được:

\(x+\sqrt{x^2+2021}=\sqrt{y^2+2021}-y\) (2)

Cộng vế với vế (1) và (2):

\(x+y+\sqrt{x^2+2021}+\sqrt{y^2+2021}=\sqrt{x^2+2021}+\sqrt{y^2+2021}-x-y\)

\(\Leftrightarrow2x+2y=0\Rightarrow A=0\)