Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Find the answer

For how many integer values of $b$ does there exist a polynomial function with integer coefficients such that $f(2)=2010$ and $f(b)=8$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

We can take $f(x)=-\frac{2002}{d}(x-b)+2010$ for all divisors $d$ of -2002. To see that we can't get any others, note that $b-2$ must divide $f(b)-f(2)$ , so $b-2$ divides -2002 (this is because $b-2$ divides $b^{n}-2^{n}$ and hence any sum of numbers of the form $b^{n}-2^{n}$ ).

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