Each of X and Y is a positive integer such that each of X+Y and X/Y is a perfect square.

Does there exist an infinite number of pairs (X,Y) satisfying all the given conditions?
Give reasons for your answer.

(In reply to Technical solution--but a larger question remains. by Charlie)

Of course there are. 
Take any perfect square that is a multiple of 5.
Make x be 4/5 and y be 1/5  of that square.
x/y will be 4.

Take any perfect square that is a multiple of (p+1) where p is a square
x = p/(p+1) and y=1/(p+1) of that square
x/y=p

There are other ratios that will also work for certain square.

Posted by Jer on 2014-09-28 10:04:46