已知1/x+1/y+1/z=1/x+y+z.求证：x、y、z中至少有两个互为相反数.

Question

已知1/x+1/y+1/z=1/x+y+z.求证：x、y、z中至少有两个互为相反数.

Answer 1

1/x+1/y+1/z=1/(x+y+z)(xy+xz+yz)/xyz=1/(x+y+z)交叉相乘------(x2y-->x^2*y)(x2y+xy2)+(x2z+xyz)+(xyz+zy2)+(xz2+yz2)=0xy(x+y)+xz(x+y)+.=0(xy+xz+yz+z2)(x+y)=0(y+z)(x+z)(x+y)=0所以x、y、z中至少有两个互为相反...