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Divisibility to square (Posted on 2011-03-05) Difficulty: 3 of 5
Each of x and y is a positive integer such that x2 + y2 + x is divisible by 2xy.

Prove that x is the square of an integer.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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re: Do any exist? | Comment 3 of 4 |
(In reply to Do any exist? by Gamer)

Nope.

After substituting x=z^2 and y=az^2 you can divide by z^2 and get

z^2(a^2 + 1) + 1 = 2kaz^2 which is possible only if z=1.

Then x=1 and y=a. Plugging in these values gives a quadratic in 'a' with discriminant 4(k^2 - 2).

But k^2 - 2 = 2 or 3 mod4, so will never be a square.

Thus, there are no solutions.


  Posted by xdog on 2011-03-09 10:51:09
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