Drop it Like it's a Floor Function

Algebra Level 5

Let nn be a positive real. If the number of real solutions to nx2=nx2+xnx^2=n\lfloor x^2\rfloor+x is 20142014, then the range of possible values of nn is n(a,b]n\in (a,b\,] If ba=Nb-a=N, then find the value of 1000N\lfloor 1000N\rfloor

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