The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs $(x,y)$ of positive integers is the harmonic mean of $x$ and $y$ equal to $20$?
+128578 Let x, y be the integers
Their reciprocals = 1/x and 1/y
The arithmetic mean of their reciprocals = [ 1/x + 1/y] / 2 = [ x + y] / [ 2xy]
The reciprocal of their arithmetic mean = [ 2xy] / [ x + y] = harmonic mean
So
2xy / [ x + y ] = 20
xy / [ x + y] = 10
xy = 10x + 10y
yx - 10y = 10x
y ( x - 10) = 10x
y = 10x / [ x - 10]
If
x = 11 y = 110
x = 12 y = 60
x = 15 y = 30
x = 20 y = 20
x = 30 y = 15 we can stop here
(x, y) = (11,110) , ( 12,60) , ( 15,30) (20, 20) ( 30, 15) (60, 12) ( 110,11)
