Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2009 AIME I Problems/Problem 6

Revision as of 22:43, 19 March 2009 by Moplam (talk | contribs) (→‎Solution)

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,

So let do case work:

For ${\lfloor x\rfloor}=0$ N=$1$ no matter what x is

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

Answer $= 1+5+37+369= \boxed {412}$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2009_AIME_I_Problems/Problem_6&oldid=30817"