Bài 1 : Cho x,y,z thoả mãn [tex]\frac{1}{x} + \frac{1}{y} + \frac{1}{z}=\frac{1}{x+y+z}[/tex] Chứng minh rằng: [tex]\frac{1}{x^{2019}} + \frac{1}{y^{2019}} + \frac{1}{z^{2019}} = \frac{1}{x^{2019}+y^{2019}+z^{2019}}[/tex] Mấy anh chị học giỏi giúp e với ạ
[tex]\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z} \Leftrightarrow \frac{1}{x+y+z}-\frac{1}{x}=\frac{y+z}{yz}[/tex] [tex]\Leftrightarrow \frac{x-x-y-z}{x(x+y+z)}=\frac{y+z}{yz} \Leftrightarrow \frac{-(y+z)}{x(x+y+z)}=\frac{y+z}{yz} \Leftrightarrow -yz(y+z)=(y+z)(x^2+xy+zx)[/tex] [tex]\Leftrightarrow (y+z)(x^2+xy+yz+zx)=0 \Leftrightarrow (x+y)(y+z)(z+x)=0[/tex] TH1: [tex]x+y=0 \Leftrightarrow x=-y \Leftrightarrow x^{2019}=-y^{2019}[/tex] [tex]\Leftrightarrow \frac{1}{x^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{-1}{y^{2019}}+\frac{1}{y^{2019}}+\frac{1}{z^{2019}}=\frac{1}{z^{2019}}=\frac{1}{z^{2019}+y^{2019}+x^{2019}}[/tex] ( đpcm ) TH2;3: tương tự.
