1: \(\Leftrightarrow a\sqrt{a}+b\sqrt{b}>=\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)\)

=>\(\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b-\sqrt{ab}\right)>=0\)

=>\(\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)^2>=0\)(luôn đúng)

Đặt \(\left\{{}\begin{matrix}\sqrt{a^2+b^2}=x\\\sqrt{b^2+c^2}=y\\\sqrt{c^2+a^2}=z\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}a^2=\dfrac{x^2+z^2-y^2}{2}\\b^2=\dfrac{x^2+y^2-z^2}{2}\\c^2=\dfrac{y^2+z^2-x^2}{2}\\x+y+z=\sqrt{2011}\end{matrix}\right.\)

Và \(\left\{{}\begin{matrix}b+c\le\sqrt{2\left(b^2+c^2\right)}=\sqrt{2}y\\a+b\le\sqrt{2}x\\c+a\le\sqrt{2}z\end{matrix}\right.\)

\(VT=\dfrac{1}{2\sqrt{2}}\left(\dfrac{x^2+z^2-y^2}{y}+\dfrac{x^2+y^2-z^2}{z}+\dfrac{y^2+z^2-x^2}{x}\right)\)

\(\ge\dfrac{1}{2\sqrt{2}}\left(\dfrac{2\left(x+y+z\right)^2}{\left(x+y+z\right)}-\left(x+y+z\right)\right)\)

\(=\dfrac{1}{2\sqrt{2}}\left(x+y+z\right)=\dfrac{\sqrt{2011}}{2\sqrt{2}}=VP\)

Từng sau em hạn chế đăng nhiều bài cùng một lúc như thế này nhé. 

Bài 1:

Ta có: \(a+\frac{4}{(a-b)(b+1)^2}=(a-b)+\frac{b+1}{2}+\frac{b+1}{2}+\frac{4}{(a-b)(b+1)^2}-1\)

Áp dụng BĐT AM-GM cho các số không âm ta có:

\((a-b)+\frac{b+1}{2}+\frac{b+1}{2}+\frac{4}{(a-b)(b+1)^2}\geq 4\sqrt[4]{\frac{4(a-b)(b+1)^2}{4(a-b)(b+1)^2}}=4\)

\(\Rightarrow a+\frac{4}{(a-b)(b+1)^2}=(a-b)+\frac{b+1}{2}+\frac{b+1}{2}+\frac{4}{(a-b)(b+1)^2}-1\geq 4-1\)

\(\Leftrightarrow a+\frac{4}{(a-b)(b+1)^2}\geq 3\)

Ta có đpcm.

Dấu bằng xảy ra khi \(a-b=\frac{b+1}{2}=\frac{4}{(a-b)(b+1)^2}\)

\(\Leftrightarrow a=2; b=1\)

Bài 2:

Đặt \(\left(\frac{a}{b}, \frac{b}{c}, \frac{c}{a}\right)\mapsto (x,y,z)\Rightarrow xyz=1\)

BĐT cần chứng minh tương đương với:

\(x^2+y^2+z^2\geq \frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

\(\Leftrightarrow x^2+y^2+z^2\geq \frac{xy+yz+xz}{xyz}=xy+yz+xz(*)\)

Áp dụng BĐT AM-GM:

\(x^2+y^2\geq 2\sqrt{x^2y^2}=2xy\)

\(y^2+z^2\geq 2\sqrt{y^2z^2}=2yz\)

\(z^2+x^2\geq 2\sqrt{z^2x^2}=2zx\)

Cộng theo vế: \(\Rightarrow 2(x^2+y^2+z^2)\geq 2(xy+yz+xz)\)

\(\Leftrightarrow x^2+y^2+z^2\geq xy+yz+xz\)

Do đó (*) đúng, ta có đpcm.

Dấu bằng xảy ra khi \(x=y=z=1\Leftrightarrow a=b=c\)

Bài 3:

Ta có: \(\text{VT}=(\frac{b}{\sqrt{a}}+\frac{c}{\sqrt{b}}+\frac{a}{\sqrt{c}})+(\frac{c}{\sqrt{a}}+\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}})\)

Áp dụng BĐT Bunhiacopxky:

\((\frac{b}{\sqrt{a}}+\frac{c}{\sqrt{b}}+\frac{a}{\sqrt{c}})(\sqrt{a}+\sqrt{b}+\sqrt{c})\geq (\sqrt{b}+\sqrt{c}+\sqrt{a})^2\)

\(\frac{b}{\sqrt{a}}+\frac{c}{\sqrt{b}}+\frac{a}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}(1)\)

Áp dụng BĐT AM-GM:

\(\frac{c}{\sqrt{a}}+\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}\geq 3\sqrt[3]{\frac{abc}{\sqrt{abc}}}=3(2)\) do $abc=1$

Từ \((1); (2)\Rightarrow \text{VT}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3\) (đpcm)

Dấu bằng xảy ra khi \(a=b=c=1\)

Từ giả thiết \(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ca}=1\Rightarrow xy+yz+xz=1\left(x=\dfrac{1}{a};y=\dfrac{1}{b};z=\dfrac{1}{c}\right)\)

\(A=\sum\dfrac{1}{\sqrt{1+a^2}}=\sum\dfrac{\dfrac{1}{a}}{\sqrt{\dfrac{1}{a^2}+1}}=\sum\dfrac{x}{\sqrt{x^2+1}}=\sum\dfrac{x}{\sqrt{x^2+xy+yz+xz}}=\sum\dfrac{x}{\sqrt{\left(x+y\right)\left(x+z\right)}}\le\dfrac{1}{2}\sum\dfrac{x}{x+y}+\dfrac{x}{x+z}=\dfrac{3}{2}\)

\(P=\sqrt{a^2+\dfrac{1}{a^2}}+\sqrt{b^2+\dfrac{1}{b^2}}+\sqrt{c^2+\dfrac{1}{c^2}}\)

\(\Leftrightarrow\sqrt{\dfrac{97}{4}}P=\sqrt{4+\dfrac{81}{4}}\sqrt{a^2+\dfrac{1}{a^2}}+\sqrt{4+\dfrac{81}{4}}\sqrt{b^2+\dfrac{1}{b^2}}+\sqrt{4+\dfrac{81}{4}}\sqrt{c^2+\dfrac{1}{c^2}}\)

\(\ge\left(2a+\dfrac{9}{2a}\right)+\left(2b+\dfrac{9}{2b}\right)+\left(2c+\dfrac{9}{2c}\right)\)

\(=2\left(a+b+c\right)+\dfrac{9}{2}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)

\(\ge4+\dfrac{9}{2}.\dfrac{9}{a+b+c}=4+\dfrac{81}{4}=\dfrac{97}{4}\)

\(\Rightarrow P\ge\sqrt{\dfrac{97}{4}}\)

PS: Lần sau chép đề cẩn thận nhé bạn.

Ta luôn có \(\left(\dfrac{1}{\sqrt{a}}-\dfrac{1}{\sqrt{b}}\right)^2\ge0\forall a;b\)

\(\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{2}{\sqrt{ab}}\)

\(\Leftrightarrow2\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\ge\dfrac{2}{\sqrt{ab}}+\dfrac{1}{a}+\dfrac{1}{b}\)

\(\Leftrightarrow\dfrac{2\left(a+b\right)}{ab}\ge\left(\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}\right)^2\)

\(\Leftrightarrow\sqrt{\dfrac{2\left(a+b\right)}{ab}}\ge\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}\)

Tương tự cho 2 BĐT còn lại rồi cộng theo vế:

\(\sqrt{2}\left(\sqrt{\dfrac{a+b}{ab}}+\sqrt{\dfrac{b+c}{bc}}+\sqrt{\dfrac{a+c}{ac}}\right)\ge2\left(\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}+\dfrac{1}{\sqrt{c}}\right)\)

\(\Leftrightarrow\sqrt{\dfrac{a+b}{ab}}+\sqrt{\dfrac{b+c}{bc}}+\sqrt{\dfrac{a+c}{ac}}\ge\sqrt{\dfrac{2}{a}}+\sqrt{\dfrac{2}{b}}+\sqrt{\dfrac{2}{c}}\)

\("="\Leftrightarrow a=b=c\)