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Mock AIME 2 2006-2007/Problem 2

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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?

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 + \f.... So in fact, there are 0 such elements of S.

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