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

\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\frac{1}{x+y+z}=0\)

\(\Leftrightarrow\frac{x+y}{xy}+\frac{x+y}{z\left(x+y+z\right)}=0\)

\(\Leftrightarrow\left(x+y\right)\cdot\frac{xy+z\left(x+y+z\right)}{xyz\left(x+y+z\right)}=0\)

\(\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=0\)

\(\Leftrightarrow x=-y\left(h\right)y=-z\left(h\right)z=-x\)

Nếu 

\(x=-y\Rightarrow\frac{1}{x^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{x^{2019}}-\frac{1}{x^{2019}}+\frac{1}{z^{2019}}=\frac{1}{z^{2019}}\)

\(\frac{1}{x^{2019}+y^{2019}+z^{2019}}=\frac{1}{x^{2019}-x^{2019}+z^{2019}}=\frac{1}{z^{2019}}\)

Tương tự các TH còn lại nha!

P/S:Có 1 bài chặt hơn ntnày:

Cho \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\) thì \(\frac{1}{x^n}+\frac{1}{y^n}+\frac{1}{z^n}=\frac{1}{x^n+y^n+z^n}\) với n lẻ.

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\)=> \(\frac{xy+yz+zx}{xyz}=\frac{1}{x+y+z}\)

=> (x+y+z)(xy+yz+zx) = xyz

=> \(x^2y+xy^2+y^2z+yz^2+zx^2+z^2x+2xyz=0\)

=> (x+y)(y+z)(z+x) = 0

=> \(\left[{}\begin{matrix}x=-y\\y=-z\\z=-x\end{matrix}\right.\)

TH1: x = -y

=> \(\frac{1}{x^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{\left(-y\right)^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{z^{2019}}\)

=> \(\frac{1}{x^{2019}+y^{2019}+z^{2019}}=\frac{1}{\left(-y\right)^{2019}+y^{2019}+z^{2019}}=\frac{1}{z^{2019}}\)

=> ĐPCM

Tương tự với TH2 và TH3

Sửa đề : \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2019}\)

Thay \(2019=x+y+z\)ta có :

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

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

\(\Leftrightarrow\frac{y}{xy}+\frac{x}{xy}=\frac{z}{z\left(x+y+z\right)}-\frac{x+y+z}{z\left(x+y+z\right)}\)

\(\Leftrightarrow\frac{x+y}{xy}=\frac{z-x-y-z}{z\left(x+y+z\right)}\)

\(\Leftrightarrow\frac{x+y}{xy}=\frac{-\left(x+y\right)}{z\left(x+y+z\right)}\)

\(\Leftrightarrow z\left(x+y\right)\left(x+y+z\right)=-xy\left(x+y\right)\)

\(\Leftrightarrow z\left(x+y\right)\left(x+y+z\right)+xy\left(x+y\right)=0\)

\(\Leftrightarrow\left(x+y\right)\left[z\left(x+y+z\right)+xy\right]=0\)

\(\Leftrightarrow\left(x+y\right)\left(xz+yz+z^2+xy\right)=0\)

\(\Leftrightarrow\left(x+y\right)\left[z\left(x+z\right)+y\left(x+z\right)\right]=0\)

\(\Leftrightarrow\left(x+y\right)\left(x+z\right)\left(y+z\right)=0\)

( mình chỉ xét 1 t/h, các t/h còn lại hoàn toàn tương tự )

TH1 : \(x+y=0\)

\(\Leftrightarrow x=-y\)(1)

Thay (1) vào A ta có :

\(A=\frac{1}{-y^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}\)

\(A=\frac{1}{z^{2019}}\)

Mặt khác : \(x+y+z=2019\)

Thay (1) vào đẳng thức trên ta được : \(-y+y+z=2019\)

\(\Leftrightarrow z=2019\)

Thay z vào A ta được : \(A=\frac{1}{2019^{2019}}\)

sửa đền nha:^{\(\frac{1}{x}\)}+\(\frac{1}{y}\)+\(\frac{1}{z}\)=\(\frac{1}{2019}\)

nguyệt dạ hình như là mai linh... t đang tìm bài này thì thấy lp mk ... thành ra trong lp hỏi nhau :)))

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

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

\(\Leftrightarrow\frac{x+y}{xy}=\frac{z}{\left(x+y+z\right).z}-\frac{x+y+z}{z.\left(x+y+z\right)}=\frac{-x-y}{z.\left(x+y+z\right)}\)

\(\Leftrightarrow\frac{x+y}{xy}=\frac{x+y}{-z.\left(x+y+z\right)}\)

TH1: x+y=0

=> x=-y => P=0

TH2: xy=-z.(x+y+z)

\(\Leftrightarrow xy=-xz-zy-z^2\Leftrightarrow xy+xz+zy+z^2=0\Leftrightarrow x.\left(y+z\right)+z.\left(y+z\right)=0\)

\(\Leftrightarrow\left(x+z\right).\left(y+z\right)=0\Leftrightarrow\orbr{\begin{cases}x=-z\\y=-z\end{cases}\Rightarrow P=0}\)