Phân số cuối cùng chắc em ghi nhầm

\(\dfrac{x}{y+z+t}+\dfrac{y+z+t}{9x}\ge2\sqrt{\dfrac{x\left(y+z+t\right)}{9x\left(y+z+t\right)}}=\dfrac{2}{3}\)

Tương tự:

\(\dfrac{y}{z+t+x}+\dfrac{z+t+x}{9y}\ge\dfrac{2}{3}\)

\(\dfrac{z}{t+x+y}+\dfrac{t+x+y}{9z}\ge\dfrac{2}{3}\)

\(\dfrac{t}{x+y+z}+\dfrac{x+y+z}{9t}\ge\dfrac{2}{3}\)

Đồng thời:

\(\dfrac{8}{9}\left(\dfrac{y+z+t}{x}+\dfrac{z+t+x}{y}+\dfrac{t+x+y}{z}+\dfrac{x+y+z}{t}\right)\)

\(\ge\dfrac{8}{9}\left(\dfrac{3\sqrt[3]{yzt}}{x}+\dfrac{3\sqrt[3]{ztx}}{y}+\dfrac{3\sqrt[3]{txy}}{z}+\dfrac{3\sqrt[3]{xyz}}{t}\right)\)

\(\ge\dfrac{8}{3}.4\sqrt[4]{\dfrac{\sqrt[3]{yzt}.\sqrt[3]{ztx}.\sqrt[3]{txy}.\sqrt[3]{xyz}}{xyzt}}=\dfrac{32}{3}\)

Cộng vế:

\(VT\ge4.\dfrac{2}{3}+\dfrac{32}{3}=\dfrac{40}{3}\)

Dấu "=" xảy ra khi \(x=y=z=t\)

\(VP=\frac{x}{y+z+t}+\frac{y}{z+t+x}+\frac{z}{t+x+y}+\frac{t}{x+y+z}+\frac{y+z+t}{x}+\frac{z+t+x}{y}+\frac{t+x+y}{z}+\frac{x+y+z}{t}=\left(\frac{x}{y+z+t}+\frac{y+z+t}{9x}\right)+\left(\frac{y}{z+t+x}+\frac{z+t+x}{9y}\right)+\left(\frac{z}{t+x+y}+\frac{t+x+y}{9z}\right)+\left(\frac{t}{x+y+z}+\frac{x+y+z}{9t}\right)+\frac{8}{9}\left(\frac{y+z+t}{x}+\frac{z+t+x}{y}+\frac{t+x+y}{z}+\frac{x+y+z}{t}\right)\)\(\ge8\sqrt[8]{\frac{x}{y+z+t}.\frac{y}{z+t+x}.\frac{z}{t+x+y}.\frac{t}{x+y+z}.\frac{y+z+t}{9x}.\frac{z+t+x}{9y}.\frac{t+x+y}{9z}.\frac{x+y+z}{9t}}+\frac{8}{9}\left(\frac{y}{x}+\frac{z}{x}+\frac{t}{x}+\frac{z}{y}+\frac{t}{y}+\frac{x}{y}+\frac{t}{z}+\frac{x}{z}+\frac{y}{z}+\frac{x}{t}+\frac{y}{t}+\frac{z}{t}\right)\)\(\ge\frac{8}{3}+\frac{8}{9}.12\sqrt[12]{\frac{y}{x}.\frac{z}{x}.\frac{t}{x}.\frac{z}{y}.\frac{t}{y}.\frac{x}{y}.\frac{t}{z}.\frac{x}{z}.\frac{y}{z}.\frac{x}{t}.\frac{y}{t}.\frac{z}{t}}=\frac{8}{3}+\frac{8}{9}.12=\frac{40}{3}=VT\left(đpcm\right)\)

Đẳng thức xảy ra khi x = y = z = t > 0 

Ko biết

b viết lại cái đề đi mik k hieuur

*)Nếu \(x=y=z=t\)
\(\dfrac{x}{y+z+t}=\dfrac{y}{z+t+x}=\dfrac{z}{y+x+t}=\dfrac{t}{x+y+z}\)
Áp dụng tích chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x}{y+z+t}=\dfrac{y}{z+t+x}=\dfrac{z}{y+x+t}=\dfrac{t}{x+y+z}=\dfrac{x+y+z+t}{3\left(x+y+z+t\right)}=\dfrac{1}{3}\)=> \(P=\dfrac{2x}{2x}+\dfrac{2x}{2x}+\dfrac{2x}{2x}+\dfrac{2x}{2x}=4\)
*)Nếu có ít nhất 2 số khác nhau , giả sử \(x\ne y\)
=> \(\dfrac{x}{y+z+t}=\dfrac{y}{x+z+t}=\dfrac{x-y}{y+z+t-x-z-t}=\dfrac{x-y}{y-x}=-1\)
=> \(x=-\left(y+z+t\right)\Rightarrow x+y+z+t=0\)
=> \(\left[{}\begin{matrix}x+y=-\left(z+t\right)\Rightarrow\dfrac{x+y}{z+t}=-1\\y+z=-\left(t+x\right)\Rightarrow\dfrac{y+z}{t+x}=-1\\z+t=-\left(x+y\right)\Rightarrow\dfrac{z+t}{x+y}=-1\\t+x=-\left(y+z\right)\Rightarrow\dfrac{t+x}{y+z}=-1\end{matrix}\right.\)
=> \(P=-1-1-1-1=-4\)
Vậy P=4
P = -4

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