Đặt \(\left(a;2b;3c\right)=\left(x;y;z\right)\Rightarrow x+y+z=3\)

\(Q=\dfrac{x+1}{1+y^2}+\dfrac{y+1}{1+z^2}+\dfrac{z+1}{1+x^2}\)

Ta có:

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

Tương tự:

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

Cộng vế:

\(Q\ge\dfrac{x+y+z}{2}+3-\dfrac{1}{2}\left(xy+yz+zx\right)\)

\(Q\ge\dfrac{x+y+z}{2}+3-\dfrac{1}{6}\left(x+y+z\right)^2=\dfrac{3}{2}+3-\dfrac{9}{6}=3\)

\(Q_{min}=3\) khi \(x=y=z=1\) hay \(\left(a;b;c\right)=\left(1;\dfrac{1}{2};\dfrac{1}{3}\right)\)

Áp dụng bất đẳng thức: \(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\)

\(\Leftrightarrow\left(a+b\right)^2\ge4ab\) \(\Leftrightarrow a^2+2ab+b^2\ge4ab\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow\left(a-b\right)^2\ge0\left(đúng\right)\)

\(\dfrac{1}{2a+b+c}=\dfrac{1}{4}.\dfrac{4}{2a+b+c}\le\dfrac{1}{4}\left(\dfrac{1}{2a}+\dfrac{1}{b+c}\right)\le\dfrac{1}{4}\left[\dfrac{1}{2a}+\dfrac{1}{4}\left(\dfrac{1}{b}+\dfrac{1}{c}\right)\right]=\dfrac{1}{8}\left(\dfrac{1}{a}+\dfrac{1}{2b}+\dfrac{1}{2c}\right)\)

CMTT \(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{a+2b+c}\le\dfrac{1}{8}\left(\dfrac{1}{2a}+\dfrac{1}{b}+\dfrac{1}{2c}\right)\\\dfrac{1}{a+b+2c}\le\dfrac{1}{8}\left(\dfrac{1}{2a}+\dfrac{1}{2b}+\dfrac{1}{c}\right)\end{matrix}\right.\)

\(\Rightarrow M=\dfrac{1}{2a+b+c}+\dfrac{1}{a+2b+c}+\dfrac{1}{a+b+2c}\le\dfrac{1}{8}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{2}{2a}+\dfrac{2}{2b}+\dfrac{2}{2c}\right)=\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)=\dfrac{1}{4}.4=1\)

\(minM=1\Leftrightarrow a=b=c=\dfrac{3}{4}\)

Sửa lại \(minM=1\rightarrow maxM=1\)

\(a^2+2b^2+3=\left(a^2+b^2\right)+\left(b^2+1\right)+2\ge2ab+2b+2=2\left(ab+b+1\right)\)

Tương tự ...

\(\Rightarrow P\le\dfrac{1}{2\left(ab+b+1\right)}+\dfrac{1}{2\left(bc+c+1\right)}+\dfrac{1}{2\left(ca+a+1\right)}\)

\(=\dfrac{1}{2}\left(\dfrac{c}{abc+bc+c}+\dfrac{1}{bc+c+1}+\dfrac{bc}{ca.bc+a.bc+bc}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{c}{1+bc+c}+\dfrac{1}{bc+c+1}+\dfrac{bc}{c+1+bc}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{c+1+bc}{1+bc+c}\right)=\dfrac{1}{2}\)

\(P_{max}=\dfrac{1}{2}\) khi \(a=b=c=1\)

Dấu BĐT bị ngược, sửa đề: \(\dfrac{1}{a^4+b^4+2ab^4}+\dfrac{1}{a^2+b^4+2a^2b^2}\le\dfrac{1}{2}\).

Đặt \(b^2=x\left(x>0\right)\Rightarrow a+x=2ax\).

Khi đó ta cần chứng minh:

\(\dfrac{1}{a^4+x^2+2ax^2}+\dfrac{1}{a^2+x^4+2a^2x}\le\dfrac{1}{2}\)

Áp dụng BĐT AM-GM:

\(\dfrac{1}{a^4+x^2+2ax^2}+\dfrac{1}{a^2+x^4+2a^2x}\)

\(\le\dfrac{1}{2a^2x+2ax^2}+\dfrac{1}{2ax^2+2a^2x}\)

\(=\dfrac{2}{2ax\left(a+x\right)}\)

\(=\dfrac{1}{ax\left(a+x\right)}\)

\(=\dfrac{1}{2a^2x^2}\)

Ta thấy: \(a+x\ge2\sqrt{ax}\)

\(\Leftrightarrow2ax\ge2\sqrt{ax}\)

\(\Leftrightarrow ax-\sqrt{ax}\ge0\)

\(\Leftrightarrow\sqrt{ax}\left(\sqrt{ax}-1\right)\ge0\)

\(\Leftrightarrow\sqrt{ax}\ge1\)

\(\Rightarrow ax\ge1\)

Khi đó: \(\dfrac{1}{2a^2x^2}\le\dfrac{1}{2}\)

\(\Rightarrow\dfrac{1}{a^4+x^2+2ax^2}+\dfrac{1}{a^2+x^4+2a^2x}\le\dfrac{1}{2}\)

Hay \(\dfrac{1}{a^4+b^4+2ab^4}+\dfrac{1}{a^2+b^4+2a^2b^2}\le\dfrac{1}{2}\).

\(\dfrac{1}{\left(a+b+a+c\right)^2}\le\dfrac{1}{4\left(a+b\right)\left(a+c\right)}=\dfrac{1}{4\left(a^2+ab+bc+ca\right)}\le\dfrac{1}{64}\left(\dfrac{1}{a^2}+\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ca}\right)\)

\(\le\dfrac{1}{64}\left(\dfrac{1}{a^2}+\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=\dfrac{1}{64}\left(\dfrac{2}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)\)

Tương tự và cộng lại:

\(P\le\dfrac{1}{64}\left(\dfrac{4}{a^2}+\dfrac{4}{b^2}+\dfrac{4}{c^2}\right)=\dfrac{1}{16}.3=\dfrac{3}{16}\)

Dấu "=" xảy ra khi \(a=b=c=1\)

Áp dụng bđt: \(\dfrac{1}{x+y}\le\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\left(1\right)\)

\(\dfrac{1}{2a+b+c}=\dfrac{1}{\left(a+b\right)+\left(a+c\right)}\le\dfrac{1}{4}\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)\)

\(\Rightarrow P\le\dfrac{1}{16}\left[\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)^2+\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}\right)^2+\left(\dfrac{1}{b+c}+\dfrac{1}{a+c}\right)^2\right]\)\(\Rightarrow16P\le\dfrac{2}{\left(a+b\right)^2}+\dfrac{2}{\left(b+c\right)^2}+\dfrac{2}{\left(a+c\right)^2}+\dfrac{2}{\left(a+b\right)\left(b+c\right)}+\dfrac{2}{\left(a+b\right)\left(b+c\right)}+\dfrac{2}{\left(b+c\right)\left(c+a\right)}\)

Áp dụng: \(x^2+y^2+z^2\ge xy+yz+xz\left(2\right)\) với a+b=x,b+c=y,c+a=z

\(\Rightarrow16P\le\dfrac{4}{\left(a+b\right)^2}+\dfrac{4}{\left(b+c\right)^2}+\dfrac{4}{\left(c+a\right)^2}\)

Ta có: \(\dfrac{1}{\left(a+b\right)^2}\le4.16.\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2\)(do (1))

\(\Rightarrow16P\le\dfrac{1}{4}.16\left[\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2+\left(\dfrac{1}{b}+\dfrac{1}{c}\right)^2+\left(\dfrac{1}{c}+\dfrac{1}{a}\right)^2\right]=\dfrac{1}{4}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ca}\right)\le\dfrac{1}{4}.4.\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=3\)(do(2) và \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=3\))

\(\Rightarrow P\le\dfrac{3}{16}\)

\(ĐTXR\Leftrightarrow a=b=c=1\)

Đặt \(\left\{{}\begin{matrix}a-2=x\ge0\\b=y\ge0\end{matrix}\right.\) \(\Rightarrow2y+4=\left(x+2\right)y\Rightarrow xy=4\)

\(P=\dfrac{\sqrt{x^2+2x}}{x+1}+\dfrac{\sqrt{y^2+2y}}{y+1}+\dfrac{1}{x+y+2}\)

\(P=\dfrac{\sqrt{2x\left(x+2\right)}}{\sqrt{2}\left(x+1\right)}+\dfrac{\sqrt{2y\left(y+2\right)}}{\sqrt{2}\left(y+1\right)}+\dfrac{1}{x+1+y+1}\)

\(P\le\dfrac{1}{2\sqrt{2}}\left(\dfrac{3x+2}{x+1}+\dfrac{3y+2}{y+1}\right)+\dfrac{1}{4}\left(\dfrac{1}{x+1}+\dfrac{1}{y+1}\right)\)

\(P\le\dfrac{1}{2\sqrt{2}}\left(3-\dfrac{1}{x+1}+3-\dfrac{1}{y+1}\right)+\dfrac{1}{4}\left(\dfrac{1}{x+1}+\dfrac{1}{y+1}\right)\)

\(P\le\dfrac{3\sqrt{2}}{2}-\dfrac{\sqrt{2}-1}{4}\left(\dfrac{1}{x+1}+\dfrac{1}{y+1}\right)\)

Ta có:

\(\dfrac{1}{x+1}+\dfrac{1}{y+1}=\dfrac{x+y+2}{xy+x+y+1}=\dfrac{x+y+2}{x+y+5}=1-\dfrac{3}{x+y+5}\ge1-\dfrac{3}{2\sqrt{xy}+5}=\dfrac{2}{3}\)

\(\Rightarrow P\le\dfrac{3\sqrt{3}}{2}-\dfrac{\sqrt{2}-1}{4}.\dfrac{2}{3}=...\)

Dấu "=" xảy ra khi \(x=y=2\) hay \(\left(a;b\right)=\left(4;2\right)\)

\(2ab+a+b=2a^2+2b^2\ge2ab+\dfrac{1}{2}\left(a+b\right)^2\Rightarrow a+b\le2\)

\(F=\dfrac{a^4}{ab}+\dfrac{b^4}{ab}+2020\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\ge\dfrac{\left(a^2+b^2\right)^2}{2ab}+\dfrac{8080}{a+b}\ge a^2+b^2+\dfrac{8080}{a+b}\)

\(F\ge\dfrac{\left(a+b\right)^2}{2}+\dfrac{8080}{a+b}=\dfrac{\left(a+b\right)^2}{2}+\dfrac{4}{a+b}+\dfrac{4}{a+b}+\dfrac{8072}{a+b}\)

\(F\ge3\sqrt[3]{\dfrac{16\left(a+b\right)^2}{\left(a+b\right)^2}}+\dfrac{8072}{2}=...\)