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.
(In reply to Possible Solution
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).