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9 tháng 10 2020

Ta có: \(x+\frac{1}{y}=y+\frac{1}{z}\Rightarrow x-y=\frac{1}{z}-\frac{1}{y}=\frac{y-z}{yz}\)(1)

\(y+\frac{1}{z}=z+\frac{1}{x}\Rightarrow y-z=\frac{1}{x}-\frac{1}{z}=\frac{z-x}{zx}\)(2)

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

Nhân vế theo vế ba đẳng thức (1), (2), (3), ta được: \(\left(x-y\right)\left(y-z\right)\left(z-x\right)=\frac{\left(x-y\right)\left(y-z\right)\left(z-x\right)}{x^2y^2z^2}\)

\(\Rightarrow\orbr{\begin{cases}\left(x-y\right)\left(y-z\right)\left(z-x\right)=0\left(^∗\right)\\x^2y^2z^2=1\end{cases}}\)

Từ (*) ta giả sử x - y = 0 thì x = y khi đó \(\frac{1}{y}=\frac{1}{z}\Rightarrow y=z\)suy ra x = y = z. Tương tự đối với y - z = 0; z - x = 0

Vậy x = y = z hoặc x2y2z2 = 1

19 tháng 3 2020

\(x+\frac{1}{y}=y+\frac{1}{z}\Rightarrow x-y=\frac{1}{z}-\frac{1}{y}=\frac{z-y}{zy}\)

\(y+\frac{1}{z}=z+\frac{1}{x}\Rightarrow y-z=\frac{1}{x}-\frac{1}{z}=\frac{z-x}{xz}\)

\(z+\frac{1}{x}=x+\frac{1}{y}\Rightarrow z-x=\frac{1}{y}-\frac{1}{x}=\frac{x-y}{xy}\)

\(\Rightarrow\left(x-y\right)\left(y-z\right)\left(z-x\right)=\frac{y-z}{zy}\cdot\frac{z-x}{zx}\cdot\frac{x-y}{xy}\)

\(\Rightarrow\left(x-y\right)\left(y-z\right)\left(z-x\right)=\frac{\left(y-z\right)\left(z-x\right)\left(x-y\right)}{x^2y^2z^2}\)

\(\Rightarrow x^2y^2z^2\left(x-y\right)\left(y-z\right)\left(z-x\right)=\left(x-y\right)\left(y-z\right)\left(z-x\right)\)

\(\Rightarrow\left(x^2y^2z^2-1\right)\left(x-y\right)\left(y-z\right)\left(z-x\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x^2y^2z^2-1=0\\\left(x-y\right)\left(y-z\right)\left(z-x\right)=0\end{cases}}\)

\(\Rightarrow\orbr{\begin{cases}x^2y^2z^2=1\\x=y=z\end{cases}}\)

30 tháng 1 2017

x+1/y=y+1/z => x-y=1/z-1/y=(y-z)/yz 

Tương tự y-z=(z-x)/zx ; z-x=(x-y)/xy

Nhân theo vế các đẳng thức trên  ta đc:

(x-y)(y-z)(z-x)=(x-y)(y-z)(z-x)/x2y2z2

=>(x-y)(y-z)(z-x)x2y2z2-(x-y)(y-z)(z-x)=0

=>(x-y)(y-z)(z-x)(x2y2z2-1)=0

=>x-y=0 hoặc y-z=0 hoặc z-x=0 hoặc x2y2z2-1=0

=>x=y=z hoặc x2y2z2=1(đfcm)

31 tháng 1 2017
Bài làm mắc hai lỗi nghiêm trọng:
23 tháng 3 2020

Đặt \(a=2x+y+z;b=2y+z+x;c=2z+x+y\)

\( \implies\) \(a+b+c=\left(2x+y+z\right)+\left(2y+z+x\right)+\left(2z+x+y\right)\) 

\( \implies\) \(a+b+c=4x+4y+4z\)

\( \implies\) \(x+y+z=\frac{a+b+c}{4}\) 

+)Ta có : \(a=2x+y+z\)

\(\iff\) \(a=x+\left(x+y+z\right)\)

\(\iff\) \(a-\left(x+y+z\right)=x\)

\(\iff\) \(a-\frac{a+b+c}{4}=x\)

\(\iff\) \(x=\frac{3a-b-c}{4}\)

+)Ta có :\(b=2y+z+x\)

\(\iff\) \(b=y+\left(y+z+x\right)\)

\(\iff\)\(b-\left(y+z+x\right)=y\)

\(\iff\) \(b-\frac{a+b+c}{4}=y\)

\(\iff\)\(y=\frac{3b-c-a}{4}\)

+)Ta có :\(c=2z+x+y\)

\(\iff\) \(c=z+\left(z+x+y\right)\)

\(\iff\) \(c-\left(z+x+y\right)=z\)

\(\iff\) \(c-\frac{a+b+c}{4}=z\)

\(\iff\)\(z=\frac{3c-a-b}{4}\)

​​\( \implies\)​ \(\frac{x}{2x+y+z}+\frac{y}{2y+z+x}+\frac{z}{2z+x+y}\) 

 \(=\frac{3a-b-c}{4a}+\frac{3b-c-a}{4b}+\frac{3c-a-b}{4c}\)

 \(=\frac{9}{4}-\left(\frac{b}{4a}+\frac{c}{4a}+\frac{c}{4b}+\frac{a}{4b}+\frac{a}{4c}+\frac{b}{4c}\right)\)

 \(=\frac{9}{4}-\frac{1}{4}\left(\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}\right)\)

 \(=\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\)

Áp dụng bất đẳng thức ( BĐT Cosi ) : \(m+n\)\( \geq\)\(2\sqrt{mn}\) \(\left(m;n>0\right)\)ta được : 

\(\frac{b}{a}+\frac{a}{b}\) \( \geq\) 2 \(\sqrt{\frac{b}{a}.\frac{a}{b}}\) = 2 \( \implies\) \(\frac{b}{a}+\frac{a}{b}\) \( \geq\) 2 

\(\frac{c}{a}+\frac{a}{c}\) \( \geq\) 2 \(\sqrt{\frac{c}{a}.\frac{a}{c}}\) = 2 \( \implies\) \(\frac{c}{a}+\frac{a}{c}\) \( \geq\) 2 

\(\frac{b}{c}+\frac{c}{b}\) \( \geq\) 2 \(\sqrt{\frac{b}{c}.\frac{c}{b}}\) = 2 \( \implies\) \(\frac{b}{c}+\frac{c}{b}\) \( \geq\) 2 

\( \implies\) \(\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\) \( \geq\) 2 + 2 + 2 

\( \implies\) ​​\(\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\)​ \( \geq\) 6 

\( \implies\) \(\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \( \geq\) \(\frac{6}{4}\)

\( \implies\) \(\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \( \geq\) \(\frac{3}{2}\)

\( \implies\) \(-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(-\frac{3}{2}\)

\( \implies\) \(\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(\frac{9}{4}-\frac{3}{2}\)

\( \implies\) \(\frac{9}{4}-\frac{1}{4}\left[\left(\frac{b}{a}+\frac{a}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\right]\) \(\leq\) \(\frac{3}{4}\) 

23 tháng 3 2020

Dấu " = " xảy ra khi a = b = c hay x = y = z