2012 AIME II Problem 10

Find the number of positive integers n less than 1000 for which there exists a positive real number x such that n=x\lfloor x \rfloor.

Note: \lfloor x \rfloor is the greatest integer less than or equal to x.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.