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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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Hints/Tips re: Possible Solution | Comment 2 of 3 |
(In reply to Possible Solution by broll)

Lets take k(y-1)^2 = y^2+1 and do some more manipulation and solve for k and simplify: k = 1 + 2y/(y-1)^2


But for k to be a positive integer we must necessarily have 2y>=(y-1)^2.  Then 0 >= y^2-4y+1.  This inequality is true on the interval [2-sqrt(3),2+sqrt(3)].

The only integers in that interval are y=1, y=2, and y=3. Substituing back shows only y=2 yields an integer k, namely k=5.  Then the only solution to the original statement is (x,y)=(10,2).

  Posted by Brian Smith on 2022-09-15 10:40:47
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