"Suppose we arbitrarily choose 5 integer points in a plane. Show that we can always find 2 among these 5 integer points such that the line segment joining the

"Suppose we arbitrarily choose 5 integer points in a plane. Show that we can always find 2 among these 5 integer points such that the line segment joining the 2 points contains at least 1 more integer point." I am not sure what you mean. I have 'arbitrarily' picked 5 integer points on a plane. Yes, they happen to lie on line. All their line segments contain only these integer points. Perhaps you mean that is line not on "a plane"? In that case (0,0), (0,1), (0,2), (0,3), (0,1) is another counter example.