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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 $20$?

 Jun 20, 2019
 #1
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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)

 

 

cool cool cool

 Jun 20, 2019
edited by CPhill  Jun 20, 2019

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