Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Em ghi vội nó hơi sai
\(\sqrt{2012a+\frac{\left(b-c\right)^2}{2}}+\sqrt{2012b+\frac{\left(c-a\right)^2}{2}}+\sqrt{2012c+\frac{\left(a-b\right)^2}{2}}\le2012\sqrt{2}\)
\(\sqrt{2012a+\frac{\left(b-c\right)^2}{2}}=\sqrt{2a\left(a+b+c\right)+\frac{\left(b-c\right)^2}{2}}\)
\(=\sqrt{\frac{4a^2+4ab+4ac+b^2+c^2-2bc}{2}}=\sqrt{\frac{\left(2a+b+c\right)^2-4bc}{2}}\le\sqrt{\frac{\left(2a+b+c\right)^2}{2}}=\frac{1}{\sqrt{2}}\left(2a+b+c\right)\)
Tương tự:
\(\sqrt{2012b+\frac{\left(c-a\right)^2}{2}}\le\frac{1}{\sqrt{2}}\left(a+2b+c\right)\) ; \(\sqrt{2012c+\frac{\left(a-b\right)^2}{2}}\le\frac{1}{\sqrt{2}}\left(a+b+2c\right)\)
Cộng vế với vế:
\(VT\le\frac{1}{\sqrt{2}}\left(4a+4b+4c\right)=2\sqrt{2}\left(a+b+c\right)=2012\sqrt{2}\)
Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(1006;0;0\right)\) và hoán vị
Ta có \(\sqrt{2022a+\dfrac{\left(b-c\right)^2}{2}}\)
\(=\sqrt{2a\left(a+b+c\right)+\dfrac{b^2-2bc+c^2}{2}}\)
\(=\sqrt{\dfrac{4a^2+b^2+c^2+4ab+4ac-2bc}{2}}\)
\(=\sqrt{\dfrac{\left(2a+b+c\right)^2-4bc}{2}}\)
\(\le\sqrt{\dfrac{\left(2a+b+c\right)^2}{2}}\)
\(=\dfrac{2a+b+c}{\sqrt{2}}\).
Vậy \(\sqrt{2022a+\dfrac{\left(b-c\right)^2}{2}}\le\dfrac{2a+b+c}{\sqrt{2}}\). Lập 2 BĐT tương tự rồi cộng vế, ta được \(VT\le\dfrac{2a+b+c+2b+c+a+2c+a+b}{\sqrt{2}}\)
\(=\dfrac{4\left(a+b+c\right)}{\sqrt{2}}\) \(=\dfrac{4.1011}{\sqrt{2}}\) \(=2022\sqrt{2}\)
ĐTXR \(\Leftrightarrow\) \(\left\{{}\begin{matrix}ab=0\\bc=0\\ca=0\\a+b+c=1011\end{matrix}\right.\) \(\Leftrightarrow\left(a;b;c\right)=\left(1011;0;0\right)\) hoặc các hoán vị. Vậy ta có đpcm.
Gọi VT là P
Ta có:
\(\sqrt{2012a+\dfrac{\left(b-c\right)^2}{2}}=\sqrt{2a\left(a+b+c\right)+\dfrac{\left(b-c\right)^2}{2}}=\sqrt{\dfrac{\left(2a+b+c\right)^2-4bc}{2}}\le\dfrac{2a+b+c}{\sqrt{2}}\left(1\right)\)
Tương tự ta có:
\(\left\{{}\begin{matrix}\sqrt{2012b+\dfrac{\left(c-a\right)^2}{2}}\le\dfrac{2b+c+a}{\sqrt{2}}\left(2\right)\\\sqrt{2012c+\dfrac{\left(a-b\right)^2}{2}}\le\dfrac{2c+a+b}{\sqrt{2}}\left(3\right)\end{matrix}\right.\)
Cộng (1), (2), (3) vế theo vế ta được
\(P\le\dfrac{2a+b+c}{\sqrt{2}}+\dfrac{2b+c+a}{\sqrt{2}}+\dfrac{2c+a+b}{\sqrt{2}}\)
\(=\dfrac{4}{\sqrt{2}}\left(a+b+c\right)=2012\sqrt{2}\)
Dấu = xảy ra khi \(\left(a,b,c\right)=\left(1006,0,0;0,1006,0;0,0,1006\right)\)