`(x+y)/z+(y+z)/x+(x+z)/y-(x^3+y^3+z^3)/(xyz)=2`
Tính `T` với `n` là số tự nhiên.
`T=[(x+y)^n-z^n][(y+z)^n-x^n][(z+x)^n-y^n]`
`(x+y)/z+(y+z)/x+(x+z)/y-(x^3+y^3+z^3)/(xyz)=2`
`(xy(x+y)+yz(y+z)+xz(x+z))/(xyz)-(x^3+y^3+z^3)/(xyz)=2`
`(xy(x+y)+yz(y+z)+xz(x+z)-x^3-y^3-z^3)/(xyz)=2`
`x^2y+xy^2+y^2z+yz^2+x^2z+xz^2-x^3-y^3-z^3=2xyz`
`x^2y+xy^2+y^2z+yz^2+x^2z+xz^2-x^3-y^3-z^3-2xyz=0`
`(-x^3+2x^2z-xz^2+xy^2)+(x^2y-2xyz+yz^2-y^3)+(-x^2z+2xz^2-z^3+y^2z)=0`
`-x(x^2-2xz+z^2-y^2)+y(x^2-2xz+z^2-y^2)-z(x^2-2xz+z^2-y^2)=0`
`(x^2-2xz+z^2-y^2)(-x+y-z)=0`
`-[(x-z)^2-y^2](x-y+z)=0`
`-(x-z-y)(x-z+y)(x-y+z)=0`
`(x-z-y)(x-z+y)(x-y+z)=0`
`TH1:x-z-y=0`
`=>x-(y+z)=0`
`=>x=y+z` Ta có:
`T=[(x+y)^n-z^n][(y+z)^n-x^n][(z+x)^n-y^n]`
`T=[(x+y)^n-z^n][x^n-x^n][(z+x)^n-y^n]`
`T=0`
`TH2:x-z+y=0`
`=>x+y=z`
`T=[(x+y)^n-z^n][(y+z)^n-x^n][(z+x)^n-y^n]`
`T=[z^n-z^n][(y+z)^n-x^n][(z+x)^n-y^n]`
`T=0`
`TH3:x-y+z=0`
`=>z+x=y`
`T=[(x+y)^n-z^n][(y+z)^n-x^n][(z+x)^n-y^n]`
`T=[(x+y)^n-z^n][(y+z)^n-x^n][y^n-y^n]`
`T=0`
Vậy `T=0.`
