đặt a = 2x+y+z ; b = 2y+z+x ; c = 2z+x+y => a+b+c = 4x+4y+4z 
=> a - (a+b+c)/4 = x => x = (3a-b-c)/4 ; tương tự y = (3b-c-a)/4 ; z = (3c-a-b)/4 
thay vào vế trái ta có 
P = (3a-b-c)/4a + (3b-c-a)/4b + (3c-a-b)/4c = 
= 9/4 - (b/4a + c/4a + c/4b + a/4b + a/4c + b/4c) 
= 9/4 - (1/4)(b/a+a/b + c/a+a/c + c/b+b/c) 

Côsi cho từng cặp ta có: b/a+a/b ≥ 2 ; c/a+a/c ≥ 2 ; c/b+b/c ≥ 2 
=> b/a+a/b + c/a+a/c + c/b+b/c ≥ 6 
=> -(1/4)(b/a+a/b +c/a+a/c + c/b+b/c) ≤ -6/4 thay vào P ta có: 
P ≤ 9/4 - 6/4 = 3/4 (đpcm) ; dấu "=" khi a = b = c hay x = y = z 
cách này tuy biến đổi dài nhưng dễ hiểu) 
------------ 
Cách khác: 
P = x/(2x+y+z) -1 + y/(2y+z+x) -1 + z/(2z+x+y) - 1 + 3 
= -(x+y+z)/(2x+y+z) -(x+y+z)/(2y+z+x) -(x+y+z)/(2z+x+y) + 3 
= -(x+y+z).[1/(2x+y+z) + 1/(2y+z+x) + 1/(2z+x+y)] + 3 
- - - 
Côsi cho 3 số: 
2x+y+z + 2y+z+x + 2z+x+y ≥ 3.³√(2x+y+z)(2y+z+x)(2z+x+y) 
=> 4(x+y+z) ≥ 3.³√(2x+y+z)(2y+z+x)(2z+x+y) (1*) 
Côsi cho 3 số: 
1/(2x+y+z)+1/(2y+z+x)+1/(2z+x+y) ≥ 3³√1/(2x+y+z)(2y+z+x)(2z+x+y) (2*) 

Lấy (1*) *(2*) ta có: 
4(x+y+z)[1/(2x+y+z) + 1/(2y+z+x) + 1/(2z+x+y)] ≥ 9 

=> -(x+y+z).[1/(2x+y+z) + 1/(2y+z+x) + 1/(2z+x+y)] ≤ -9/4 
thay vào P ta có: 
P ≤ -9/4 + 3 = 3/4 (đpcm) ; dấu "=" khi x = y = z 

\(\left\{{}\begin{matrix}x;y;z\ge0\\x+y+z=1\end{matrix}\right.\) \(\Rightarrow0\le x;y;z\le1\)

\(\Rightarrow\left\{{}\begin{matrix}x^2\le x\\y^2\le y\\z^2\le z\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}2x^2+x+1\le x^2+2x+1\\2y^2+y+1\le y^2+2y+1\\2z^2+z+1\le z^2+2z+1\end{matrix}\right.\)

\(\Rightarrow P\le\sqrt{\left(x+1\right)^2}+\sqrt{\left(y+1\right)^2}+\sqrt{\left(z+1\right)^2}=x+y+z+3=4\)

\(P_{max}=4\) khi \(\left(x;y;z\right)=\left(0;0;1\right)\) và các hoán vị

Ta cần chứng minh: 

\(\dfrac{1}{a+b}\le\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\left(1\right)\left(a,b>0\right)\)

\(\Leftrightarrow\dfrac{4}{a+b}\le\dfrac{a+b}{ab}\\ \Leftrightarrow4ab\le\left(a+b\right)^2\\ \Leftrightarrow\left(a-b\right)^2\ge0\left(luôn.đúng\right)\)

\(DBXR\Leftrightarrow a=b\)

Do các phép biến đổi tương đương nên (1) luôn đúng

Áp dụng (1), ta có:

\(\dfrac{1}{2x+y+z}\le\dfrac{1}{4}\left(\dfrac{1}{x+y}+\dfrac{1}{x+z}\right)\le\dfrac{1}{4}\left[\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)+\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{z}\right)\right]=\dfrac{1}{16}\left(\dfrac{2}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)

Chứng minh tương tự, ta được:

\(\dfrac{1}{x+2y+z}\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{1}{z}\right)\)

\(\dfrac{1}{x+y+2z}\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{2}{z}\right)\)

Cộng từng vế BĐT, ta được:

\(\dfrac{1}{2x+y+z}+\dfrac{1}{x+2y+z}+\dfrac{1}{x+y+2z}\le\dfrac{1}{16}.\left(\dfrac{4}{x}+\dfrac{4}{y}+\dfrac{4}{z}\right)=\dfrac{1}{4}.\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)=\dfrac{1}{4}.4=1\)Hay \(\dfrac{1}{2x+y+z}+\dfrac{1}{x+2y+z}+\dfrac{1}{x+y+2z}\le1\left(đpcm\right)\)

\(DBXR\Leftrightarrow x=y=z=\dfrac{3}{4}\)

 đặt\(A=\dfrac{x^3}{2x+3y+5z}+\dfrac{y^3}{2y+3z+5x}+\dfrac{z^3}{2z+3x+5y}\)

\(=>A=\dfrac{x^4}{2x^2+3xy+5xz}+\dfrac{y^4}{2y^2+3yz+5xy}+\dfrac{z^4}{2z^2+3xz+5yz}\)

BBDT AM-GM 

\(=>A\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{2\left(x^2+y^2+z^2\right)+8\left(xy+yz+xz\right)}\)

theo BDT AM -GM ta chứng minh được \(xy+yz+xz\le x^2+y^2+z^2\)

vì \(x^2+y^2\ge2xy\)

\(y^2+z^2\ge2yz\)

\(x^2+z^2\ge2xz\)

\(=>2\left(x^2+y^2+z^2\right)\ge2\left(xy+yz+xz\right)< =>xy+yz+xz\le x^2+y^2+z^2\)

\(=>2\left(x^2+y^2+z^2\right)+8\left(xy+yz+xz\right)\le10\left(x^2+y^2+z^2\right)\)

\(=>A\ge\dfrac{\left(x^2+y^2+z^2\right)^2}{10\left(x^2+y^2+z^2\right)}=\dfrac{x^2+y^2+z^2}{10}=\dfrac{\dfrac{1}{3}}{10}=\dfrac{1}{30}\left(đpcm\right)\)

dấu"=" xảy ra<=>x=y=z=1/3

Áp dụng BĐT BSC:

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

\(\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)+\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{y}+\dfrac{1}{z}\right)+\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{z}\right)\)

\(=\dfrac{1}{16}\left(\dfrac{4}{x}+\dfrac{4}{y}+\dfrac{4}{z}\right)=\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)=\dfrac{1}{4}.4=1\)

\(maxF=1\Leftrightarrow x=y=z=\dfrac{3}{4}\)

ĐKXĐ : \(x>\frac{1}{2};y>\frac{1}{2};z>\frac{1}{2}\)

Áp dụng ( a+b)² \(\ge4ab\)ta có : 

( x+ 2y)² = \(\left(\frac{2x+y}{2}+\frac{3y}{2}\right)^2\ge4.\left(\frac{2x+y}{2}\right).\frac{3y}{2}\)

\(\Rightarrow\left(x+2y\right)^2\ge3y\left(2x+y\right)\)

\(\Rightarrow\frac{2x+y}{x+2y}\le\frac{x+2y}{3y}\)

\(\Rightarrow\frac{2x+y}{x\left(x+2y\right)}\le\frac{1}{3}\left(\frac{2}{x}+\frac{1}{y}\right)\)

Tương tự : \(\frac{2y+z}{y\left(y+2\right)}\le\frac{1}{3}\left(\frac{2}{y}+\frac{1}{z}\right)\)

                        \(\frac{2z+x}{z.\left(z+2x\right)}\le\frac{1}{3}\left(\frac{2}{z}+\frac{1}{x}\right)\)

=> \(A\le\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

Ta có : \(\sqrt{\left(2x-1\right)1}\le\frac{2x-1+1}{2}\)

\(\Rightarrow\sqrt{2x-1}\le x\)

\(\Rightarrow\frac{1}{x}\le\frac{1}{\sqrt{2x-1}}\)

        \(\frac{1}{y}\le\frac{1}{\sqrt{2y-1}}\)

           \(\frac{1}{z}\le\frac{1}{\sqrt{2z-1}}\)

Do đó 

A \(\le\frac{1}{\sqrt{2x-1}}+\frac{1}{\sqrt{2y-1}}+\frac{1}{\sqrt{2z-1}}\)

Vậy Max A = 3 khi x = y = z = 1

Theo Cô-si ta có:

\(3=\frac{1}{\sqrt{2x-1}}+\frac{1}{\sqrt{2y-1}}+\frac{1}{\sqrt{2z-1}}\ge\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)

\(\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\le3\)

Xét:

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\Sigma_{cyc}\frac{2x+y}{x\left(x+2y\right)}=\frac{1}{3}\left[\frac{\left(x-y\right)^2}{xy\left(x+2y\right)}+\frac{\left(y-z\right)^2}{yz\left(y+2z\right)}+\frac{\left(z-x\right)^2}{zx\left(z+2x\right)}\right]\ge0\)

\(\Rightarrow\Sigma_{cyc}\frac{2x+y}{x\left(x+2y\right)}\le3\)

\(\dfrac{x^3}{y+2z}+\dfrac{y^3}{z+2x}+\dfrac{z^3}{x+2y}=\dfrac{x^4}{xy+2xz}+\dfrac{y^4}{yz+2xy}+\dfrac{z^4}{xz+2yz}\)

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

Dấu "=" xảy ra khi \(x=y=z=\dfrac{1}{\sqrt{3}}\)