Since x2−y2=20 and x,y,z are positive integers, (x+y)*(x−y)=20, Hence x−y,x+y are factors of 20. Since x,y are positive integers, x+y is always positive, and for the product of (x+y)*(x−y) to be positive x−y must be positive. x,y are positive integers and x−y is positive x must be greater than y. The possible cases are : (x+y=10,x−y=2),(x+y=5,x−y=4) The second case fails because we get x=
9
2
,y=
1
2
but x,y are integral values For case one x=6,y=4 y3−2x2−4z≥−12 Substituting the values of x and y, we have : 64−72−4×z≥−21 −8−4×z≥−12 z≤1 Since x,y,z are positive integers, the only possible value for z is 1.