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 $x$ .)

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

Now, ${\lfloor x\rfloor}$ must be an integer, so lets do case work:

For ${\lfloor x\rfloor}=1$ , $N$ can be anything between $1^1$ to $2^1$ excluding $2^1$

This gives us $2^1-1^1=1$ $N$ 's

For ${\lfloor x\rfloor}=2$ , $N$ can be anything between $2^2$ to $3^2$ excluding $3^2$

This gives us $3^2-2^2=5$ $N$ 's

For ${\lfloor x\rfloor}=3$ , $N$ can be anything between $3^3$ to $4^3$ excluding $4^3$

This gives us $4^3-3^3=37$ $N$ 's

For ${\lfloor x\rfloor}=4$ , $N$ can be anything between $4^4$ to $5^4$ excluding $5^4$

This gives us $5^4-4^4=369$ $N$ 's

Since $x$ must be less than $5$ , we can stop here and the answer answer is $1+5+37+369= \boxed {412}$ .