Let (xi; yi), i = 1; 2; 3; 4; 5 be a set of ve distinct points with integer coordinates in the xy plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates.
The midpoint of the line connecting has integer coordinates if and only if (1) have the same parity (i.e., both are even or both odd), and (2) have the same parity.
There are only 4 possibilities for the parity pair of : (even,even), (even,odd), (odd, even), or (odd,odd). Since there are only 4 possibilities and you have 5 points, by the pigeonhole principle at least two points must have the same parity pair The line connecting these two points has a midpoint with integer coordinates.