It should be noted that x, y must both be larger than 20
Let 20 = z , x = z + a and y = z + b
So we have this
1/ ( z + a) + 1/(z + b) = 1 / z
(z + b) + ( z + a) 1
_________________ = ____
z^2 + az + bz + ab z
2z + a + b 1
_______________ = ____ cross-multiply
z^2 + az + bz +ab z
2z^2 + az + bz = z^2 + az + bz + ab
z^2 = ab
So 400 = ab
Smallest two factors of 400 added together is 25 and 16
The smallest possible value for x + y is 20 + 25 = 45 and 20 + 16 = 36
So....the smallest sum for x + y = 45 + 36 = 81