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Find N when x has a solution

Source: 2009 AIME I #6

March 18, 2009
floor functionAMCnumber theory

Problem Statement

How many positive integers N N less than 1000 1000 are there such that the equation x^{\lfloor x\rfloor} \equal{} N has a solution for x x? (The notation ⌊x⌋ \lfloor x\rfloor denotes the greatest integer that is less than or equal to x x.)
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