\(a^2-2b+6b+b^2=-10\)

\(\Leftrightarrow a^2-2a+6b+b^2+10=0\)

\(\Leftrightarrow\left(a^2-2a+1\right)+\left(b^2+6b+9\right)=0\)

\(\Leftrightarrow\left(a-1\right)^2+\left(b+3\right)^2=0\left(1\right)\)

Vì \(\hept{\begin{cases}\left(a-1\right)^2\ge0\forall a\\\left(b+3\right)^2\ge0\forall b\end{cases}\Leftrightarrow\hept{\begin{cases}\left(a-1\right)^2=0\\\left(b+3\right)^2=0\end{cases}\Leftrightarrow}\hept{\begin{cases}a=1\\b=-3\end{cases}}}\)

\(L=\frac{x+y}{z}+1+\frac{y+z}{x}+1+\frac{x+z}{y}+1-3\)

\(=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)-3=0-3=-3\)

Lời giải:

Áp dụng BĐT Cauchy-Schwarz:

\(\frac{1}{2x+y+z}\leq \frac{1}{16}\left(\frac{1}{x}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

\(\frac{1}{x+2y+z}\leq \frac{1}{16}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}\right)\)

\(\frac{1}{x+y+2z}\leq \frac{1}{16}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{z}\right)\)

Cộng theo vế 3 BĐT trên thu được:

\(M\leq \frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{2019}{4}\)

Vậy $M_{\max}=\frac{2019}{4}$. Giá trị này đạt tại $x=y=z=\frac{3}{2019}$

Áp dụng BĐT Cauchy-Schwarz dạng phân thức cho các số không âm:

\(A=\frac{x^2}{x+2y+3x}+\frac{y^2}{y+2z+3x}+\frac{z^2}{z+2x+3y}\) \(\ge\frac{\left(x+y+z\right)^2}{6\left(x+y+z\right)}\) \(=1\)

Vậy GTNN của A =1 \(\Leftrightarrow x=y=z=2\)

Liên tục áp dụng bất đẳng thức \(\frac{1}{a+b}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}\right)\) và ta có:

\(\frac{1}{3x+3y+2x}=\frac{1}{2\left(x+y\right)+\left(x+y+2z\right)}\le\frac{1}{4}\left(\frac{1}{2\left(x+y\right)}+\frac{1}{\left(x+z\right)+\left(y+z\right)}\right)\le\frac{1}{8\left(x+y\right)}+\frac{1}{16}\left(\frac{1}{x+z}+\frac{1}{y+z}\right)\)

Chứng minh tương tự tạ có:

\(\frac{1}{3x+2y+3z}\le\frac{1}{8\left(z+x\right)}+\frac{1}{16}\left(\frac{1}{x+y}+\frac{1}{y+z}\right)\)

\(\frac{1}{2x+3y+3z}\le\frac{1}{8\left(y+z\right)}+\frac{1}{16}\left(\frac{1}{z+x}+\frac{1}{x+y}\right)\)

Suy ra \(VT\le\frac{1}{8}\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)+\frac{1}{8}\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{z+x}\right)=\frac{3}{2}\)

Dấu "=" xảy ra <=> \(x=y=z=\frac{1}{4}\)

Ta có

\(\frac{2x+y+z+t}{x}=\frac{x+2y+z+t}{y}=\frac{x+y+2z+t}{z}=\frac{x+y+z+2t}{t}\)

\(\Rightarrow1+\frac{x+y+z+t}{x}=1+\frac{x+y+z+t}{y}=1+\frac{x+y+z+t}{z}=1+\frac{x+y+z+t}{t}\)

\(\Rightarrow\frac{x+y+z+t}{x}=\frac{x+y+z+t}{y}=\frac{x+y+z+t}{z}=\frac{x+y+z+t}{t}\)

Xét 2 trường hợp

Nếu \(x+y+z+t=0\)

\(\Rightarrow\left\{\begin{matrix}x+y=-z-t\\y+z=-t-x\\t+x=-y-z\\z+t=-x-y\end{matrix}\right.\)

Ta có \(\frac{x+y}{z+t}+\frac{y+z}{t+x}+\frac{z+t}{x+y}+\frac{t+x}{y+z}\)

\(=\frac{-z-t}{z+t}+\frac{-t-x}{t+x}+\frac{-x-y}{x+y}+\frac{-y-z}{y+z}\)

\(=\left(-1\right)+\left(-1\right)+\left(-1\right)+\left(-1\right)\)

\(=\left(-4\right)\)

Nếu \(x=y=z=t\)

Ta có \(\frac{x+y}{z+t}+\frac{y+z}{t+x}+\frac{z+t}{x+y}+\frac{t+x}{y+z}\)

\(=\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}+\frac{x+x}{x+x}\)

\(=1+1+1+1\)

\(=4\)

\(\frac{1}{2x+3y+3z}=\frac{1}{\left(x+y\right)+\left(x+z\right)+\left(y+z\right)+\left(y+z\right)}\le\frac{1}{16}\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{2}{y+z}\right)\)

Tương tự:

\(\frac{1}{3x+2y+3z}\le\frac{1}{16}\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{2}{x+z}\right)\) ; \(\frac{1}{3x+3y+2z}\le\frac{1}{16}\left(\frac{2}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}\right)\)

Cộng vế với vế:

\(P\le\frac{1}{4}\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)=\frac{2017}{4}\)

Dấu "=" xảy ra khi \(x=y=z=\frac{3}{4034}\)

\(\Rightarrow x^2+xy+\frac{y^2}{3}=x^2+xz+z^2+z^2+\frac{y^2}{3}\)

\(\Rightarrow xy=xz+2z^2\)

\(\Rightarrow xy+xz=2xz+2z^2\)

\(\Rightarrow x\left(y+z\right)=2z\left(x+z\right)\)

\(\Rightarrow\frac{y+z}{x+z}=\frac{2z}{x}\Rightarrow D\)