Problem:
For how many ordered pairs (x,y) of integers is it true that 0<x<y<106 and that the arithmetic mean of x and y is exactly 2 more than the geometric mean of x and y?
Solution:
From
2x+y=2+xy
it follows that
x+y−2xy(y−x)2y−x=4=4, and =2.
Because y=(2+x)2=x+4+4x is an integer, it follows that 4x must be an integer. Consequently 16x is a perfect square, and x is an integer. From (2+x)2<106, it follows that x<998. Thus the 997 solutions are (x,y)=(n2,(n+2)2), for n=1,2,…,997.