2009 AIME I Problems/Problem 6

Problem

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

Solution

First, $x$ must be less than $5$ , since otherwise $x^{\lfloor x\rfloor}$ would be at least $3125$ which is greater than $1000$ .

Now in order for $x^{\lfloor x\rfloor}$ to be an integer, $x$ must be an integral root of an integer,

Page

Toolbox

Search

2009 AIME I Problems/Problem 6

Problem

Solution