Talk:2007 AIME II Problems/Problem 14

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

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Talk:2007 AIME II Problems/Problem 14