Talk:2007 AIME II Problems/Problem 14

Here is a completed solution to 2007AIMEII-14. Let $f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }$ . $[f\left( 0 \right) = 1 \Rightarrow a_0 = 1 ]$ . $f\left( x \right)f\left( {2x^2 } \right) = f\left( {2x^3 + x} \right) \Rightarrow \ldots \Rightarrow a_n = 1$ . $f\left( { \pm i} \right)f\left( 2 \right) = f\left( { \mp i} \right) \Rightarrow f\left( { \pm i} \right) = 0 \Rightarrow \left. {\left( {x^2 + 1} \right)} \right|f\left( x \right)$ 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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Talk:2007 AIME II Problems/Problem 14