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Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 2"
(Mock AIME II 2007 hasn't ended yet, so the solution should not be posted.)
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== Problem ==
 
== Problem ==
 
The set <math>\displaystyle S</math> consists of all integers from <math>\displaystyle 1</math> to <math>\displaystyle 2007,</math> inclusive. For how many elements <math>\displaystyle n</math> in <math>\displaystyle S</math> is <math>\displaystyle f(n) = \frac{2n^3+n^2-n-2}{n^2-1}</math> an integer?
 
The set <math>\displaystyle S</math> consists of all integers from <math>\displaystyle 1</math> to <math>\displaystyle 2007,</math> inclusive. For how many elements <math>\displaystyle n</math> in <math>\displaystyle S</math> is <math>\displaystyle f(n) = \frac{2n^3+n^2-n-2}{n^2-1}</math> an integer?
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f(n) = ((n^2-1)(2n+1) + n - 1)/(n^2-1)
 
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f(n) = (2n+1) + (n-1)/(n^2-1)
 
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f(n) = (2n+1) + 1/(n+1)
 
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1/(n+1) is not an integer for any of the specified n so no solutions and the answer is 000
 

Revision as of 13:15, 9 August 2006

Problem

The set $\displaystyle S$ consists of all integers from $\displaystyle 1$ to $\displaystyle 2007,$ inclusive. For how many elements $\displaystyle n$ in $\displaystyle S$ is $\displaystyle f(n) = \frac{2n^3+n^2-n-2}{n^2-1}$ an integer?

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