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Harmonic Harmony 3 (Posted on 2022-09-15) Difficulty: 3 of 5
Determine two positive integers x and y, with x>y, such that:
  • y divides x, and:
  • x-y is the harmonic mean of x/y and x*y.

No Solution Yet Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

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Some Thoughts Possible Solution | Comment 1 of 3
(2(ky*y)*(ky/y))/((ky*y)+(ky/y)) = (ky-y), using the 2-number version of the formula in Wikipedia.

Simplifying nicely to k(y-1)^2 = y^2+1, where 2 is an obvious solution for y if k=5, making x=10.

Then x/y= 10/2 or 5, xy= 10*2=20, x-y=8. 

And indeed the harmonic mean of 5 and 20 is 8.

  Posted by broll on 2022-09-15 07:54:04
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