Greatest integer sums

Algebra Level 2

For two random positive real numbers \( x\) and \( y\) chosen uniformly and independently from the interval \((1,1000)\), determine the probability that \[ \lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor.\]

Notation: \(\lfloor \cdot \rfloor\) denotes the floor function.

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