已知(X+Y+Z)除以Z=(X-Y+Z)除以Y=(Y+Z-X)除以X,且XYZ的乘积不为0,求(X+Y)(Y+Z)(Z+X)除以(XYZ的积)的值

Question

已知(X+Y+Z)除以Z=(X-Y+Z)除以Y=(Y+Z-X)除以X,且XYZ的乘积不为0,求(X+Y)(Y+Z)(Z+X)除以(XYZ的积)的值

Answer 1

设(x+y+z)/z=(x-y+z)/y=(y+z-x)/x=k则x+y=(k-1)zy+z=(k+1)xz+x=(k+1)y2式减3式,得（x-y）k=0若k=0,则x=y=z=0,与题意不符故x=y,z=kx=ky,代入1式,得k(k-1)=2得k=2,或k=-1(X+Y)(Y+Z)(Z+X)/(XYZ)=(k-1)(k+1)(k+1)若k=2,...