## Let(xi,yi),i = 1,2,3,4,5, be a set of five 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 joining the points (a,b) and (c,d) is ((a+c)/2,(b+d)/2). The coefficient for this midpoint will have integer coefficients if and only if both a and c, and b and d have the same parity. In other words, they are both odd, or both even. Notice that the sum of two odd …