\(\sqrt{2021x^2+2020xy+2021y^2}=\sqrt{1011\left(x^2+y^2\right)+1010\left(x+y\right)^2}\)
\(\ge\sqrt{\dfrac{1011}{2}\left(x+y\right)^2+1010\left(x+y\right)^2}=\dfrac{\sqrt{3031}}{2}\left(x+y\right)\)
Tương tự:
\(\sqrt{2021y^2+2020yz+2021z^2}\ge\dfrac{\sqrt{3031}}{2}\left(y+z\right)\)
\(\sqrt{2021z^2+2020zx+2021x^2}\ge\dfrac{\sqrt{3031}}{2}\left(x+z\right)\)
Cộng vế:
\(P\ge\sqrt{3031}\left(x+y+z\right)=2021.\sqrt{3031}\)
Dấu "=" xảy ra khi \(x=y=z=\dfrac{2021}{3}\)
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