AMC 10B 2021 Fall Problem 12

Which of the following conditions is sufficient to guarantee that integers x, y, and z satisfy the equation

x(x-y)+y(y-z)+z(z-x)=1 ?

Answer Choices
A. x>y and y=z
B. x=y-1 and y=z-1
C. x=z+1 and y=x+1
D. x=z and y-1=x
E. x+y+z=1