若x+y+z=3,求(x-1)(y-1)(z-1)/(x-1)^3+(y-1)^3+(z-1)^3的值

Question

若x+y+z=3,求(x-1)(y-1)(z-1)/(x-1)^3+(y-1)^3+(z-1)^3的值

Answer 1

设:x-1=a,y-1=b,z-1=c;得:a+b+c=(x+y+z)-3=0,所以c=-(a+b);原式=abc/(a^3+b^3+c^3)=-ab*(a+b)/[a^3+b^3-(a+b)^3]=-ab(a+b)/[-3ab(a+b)]=1/3.