Question

cho 3 số x,y,z thỏa mãn \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{x+y+z}\)

cmr \(\dfrac{1}{x^{2009}}+\dfrac{1}{y^{2009}}+\dfrac{1}{z^{2009}}=\dfrac{1}{x^{2009}+y^{2009}+z^{2009}}\)

giúp mình nha

Answer 1

1/x +1/y +1/z=1/x+y+z

<=>xy+yz+zx/xyz=1/x+y+z

<=>x^2y +xy^2+ 2xyz +y^2z +zx^2 +xyz +z^2x=0

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

<=>x^2(y+z) +x(y+z)^2 +zy(y+z)=0

<=>(y+z)( x^2 +xy +xz zy)=0

<=>(y+z)[ x(x+y) +z(x+y) ]=0

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

<=>x= -y : y= -z : z= -x

Vậy phương trình kia trở thành;

-1/y^2009 + 1/y^2009 +1/z^2009=1/ -y^2009 + y^2009 +z^2009

<=> 1/z^2009 = 1/z^2009

<=> z=z (luôn đúng)