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Talk:2007 AIME II Problems/Problem 14

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Here is a completed solution to 2007AIMEII-14. Let f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }.\[f\left( 0 \right) = 1 \Rightarrow a_0 = 1\].f\left( x \right)f\left( {2x^2 } \right) = f\left( {2x^3 + x} \right) \Rightarrow \ldots \Rightarrow a_n = 1.f\left( { \pm i} \right)f\left( 2 \right) = f\left( { \mp i} \right) \Rightarrow f\left( { \pm i} \right) = 0 \Rightarrow \le... or f\left( x \right) \equiv 1(impossible). Let f_1 \left( x \right) = \frac{{f\left( x \right)}}{{x^2 + 1}}. Then f_1 \left( x \right)f_1 \left( {2x^2 } \right) = f_1 \left( {2x^3 + x} \right) and the same thing got:\[f_1 \left( x \right) \equiv 1\] or \left. {\left( {x^2 + 1} \right)} \right|f_1 \left( x \right). Let n be an integer and \[f_n \left( x \right) = \frac{{f\left( x \right)}}{{\left( {x^2 + 1} \right)^n }}\ such that \[deg f_n \left( x \right) = 0{\text{ or }}1\.Then \[f_n \left( x \right) = 1{\rm{ or }}x + 1]\.Check if f\left( 2 \right) + f\left( 3 \right) = 125 and we can easily get n = 2 and f_n \left( x \right) = 1 and f\left( 5 \right) = \boxed{625}.

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