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angel.white · #22 November 14th, 2007, 08:28 PM

Okay, in that case I have a start, maybe someone else can pick up what I don't know.

I would say we can find every instance by the following method:
A, B = random from the set {1,2,3,...,2007}
c,d = real numbers (this includes irrational and negative)

so:
$cd=A \Longrightarrow c=\frac{A}{d}$

$c+d=B$

Substitute:
$\frac{A}{d}+d=B$

Multiply by d:
$A+d^{2}=dB$

Subtract dB
$d^{2}-dB+A=0$

By Quadratic Formula:
$d=\frac{B\pm\sqrt{B^{2}-4A}}{2}$

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Therefore there will be a real solution so long as $B^{2} -4A \geq 0$

So I don't know how to find the probability that this will be the case, but I expect if you can find the probability of this, happening, then you can subtract that from100%, and that will be your answer.