Question

求方程X+Y=XY的整数解已知XYZ=1,求X/（XY+X+1）+Y/（YZ+Y+1）+Z/（ZX+Z+1）的值.(因式

Answer 1

原方程可转化为XY-X-Y+1=1即（X-1）*(Y-1)=1,故X-1=1,Y-1=1或X-1=-1,Y-1=-1便得所求

X/（XY+X+1）+Y/（YZ+Y+1）+Z/（ZX+Z+1）=XZ/Z(XY+X+1)+XYZ/XZ(YZ+Y+1)+Z/(ZX+Z+1)=XZ/(1+XZ+Z)+1/(Z+1+XZ)+Z/(ZX+Z+1)=1

原式可变为（M-5）X>1,即X>1/(M-5)