Difference between revisions of "Talk:2007 AIME II Problems/Problem 14"
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Then <math>f_1 \left( x \right)f_1 \left( {2x^2 } \right) = f_1 \left( {2x^3 + x} \right)</math> and the same thing got:<math>\[f_1 \left( x \right) \equiv 1 | Then <math>f_1 \left( x \right)f_1 \left( {2x^2 } \right) = f_1 \left( {2x^3 + x} \right)</math> and the same thing got:<math>\[f_1 \left( x \right) \equiv 1 | ||
\]</math> or <math>\left. {\left( {x^2 + 1} \right)} \right|f_1 \left( x \right)</math>. | \]</math> or <math>\left. {\left( {x^2 + 1} \right)} \right|f_1 \left( x \right)</math>. | ||
− | Let <math>n</math> be an integer and <math>\f_n \left( x \right) = \frac{{f\left( x \right)}}{{\left( {x^2 + 1} \right)^n }} | + | Let <math>n</math> be an integer and <math>\[f_n \left( x \right) = \frac{{f\left( x \right)}}{{\left( {x^2 + 1} \right)^n }}\</math> such that <math>\[deg f_n \left( x \right) = 0{\text{ or }}1\</math>.Then <math>\[f_n \left( x \right) = 1{\rm{ or }}x + 1]\</math>.Check if <math>f\left( 2 \right) + f\left( 3 \right) = 125</math> and we can easily get <math>n = 2</math> and <math>f_n \left( x \right) = 1</math> and <math>f\left( 5 \right) = \boxed{625}</math>. |
− | \</math> such that <math>\deg f_n \left( x \right) = 0{\ | ||
− | \</math>.Then <math>\f_n \left( x \right) = 1{\rm{ or }}x + 1 | ||
− | \</math>.Check if <math>f\left( 2 \right) + f\left( 3 \right) = 125</math> and we can easily get <math>n = 2</math> and <math>f_n \left( x \right) = 1</math> and <math>f\left( 5 \right) = \boxed{625}</math>. |
Revision as of 09:25, 17 March 2008
Here is a completed solution to 2007AIMEII-14. Let .$\[f\left( 0 \right) = 1 \Rightarrow a_0 = 1 \]$ (Error compiling LaTeX. Unknown error_msg).. or (impossible). Let . Then and the same thing got:$\[f_1 \left( x \right) \equiv 1 \]$ (Error compiling LaTeX. Unknown error_msg) or . Let 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{\text{ 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 and we can easily get and and .