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.

If (x^2+y^2+x)=2kxy, k>0, then by adding 2xy-x to both sides and factoring, one gets (x+y)^2=((2k+2)y+1)x

If a prime p divides x, then it must divide the left side x^2+2xy+y^2, and so p divides y^2 and thus y.

But then this implies ((2k+2)y+1) is not divisible by p, and thus since (x+y)^2 must have an even power of p in its prime factorization, so must x.

Since this applies for every prime p dividing x, this implies x must be the square of some integer.

Posted by Gamer on 2011-03-08 03:53:11