There do not exist positive integers x,y,z satisfying 2xz=y^2 & x+z=997
Proceed by using these two equations to form a quadratic equation. Since we have two equations and three variables we cannot have a unique solution. we will get infinitely many solutions.
from first one we have z = (y^2)/(2x) plug in this in equation 2, we get
x + (y^2)/(2x) = 997
Form this quadratic and then try and reason out