Mock AIME 2 2006-2007 Problems/Problem 2

Problem

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

Solution

$f(n) = \frac{2n^3+n^2-n-2}{n^2-1} = \frac{(n - 1)(2n^2 + 3n + 2)}{(n - 1)(n + 1)} = \frac{2n^2 + 3n + 2}{n + 1} = 2n + 1 + \frac1{n+1}$ . So in fact, there are 0 such elements of $S$ .

Mock AIME 2 2006-2007 (Problems, Source)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Mock AIME 2 2006-2007 Problems/Problem 2

Problem

Solution