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Difference between revisions of "Talk:2007 AIME II Problems/Problem 14"

(New page: Here is a completed solution to 2007AIMEII-14. Let <math><cmath>f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }</cmath></math>.<math>\[f\left( 0 \right) = 1 \Rightarrow a_0 = 1 \]</m...)
 
Line 1: Line 1:
 
Here is a completed solution to 2007AIMEII-14.
 
Here is a completed solution to 2007AIMEII-14.
−
Let <math><cmath>f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }</cmath></math>.<math>\[f\left( 0 \right) = 1 \Rightarrow a_0  = 1
+
Let <math>f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }</math>.<math>\[f\left( 0 \right) = 1 \Rightarrow a_0  = 1
−
\]</math>.<math><cmath>f\left( x \right)f\left( {2x^2 } \right) = f\left( {2x^3  + x} \right) \Rightarrow  \ldots  \Rightarrow a_n  = 1</cmath></math>.<math><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></math> or <math><cmath>f\left( x \right) \equiv 1</cmath></math>(impossible).
+
\]</math>.<math>f\left( x \right)f\left( {2x^2 } \right) = f\left( {2x^3  + x} \right) \Rightarrow  \ldots  \Rightarrow a_n  = 1</math>.<math>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)</math> or <math>f\left( x \right) \equiv 1</math>(impossible).
−
Let <math><cmath>f_1 \left( x \right) = \frac{{f\left( x \right)}}{{x^2  + 1}}</cmath></math>.
+
Let <math>f_1 \left( x \right) = \frac{{f\left( x \right)}}{{x^2  + 1}}</math>.
−
Then <math><cmath>f_1 \left( x \right)f_1 \left( {2x^2 } \right) = f_1 \left( {2x^3  + x} \right)</cmath></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><cmath>\left. {\left( {x^2  + 1} \right)} \right|f_1 \left( x \right)</cmath></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{\rm{ or }}1
 
\]</math> such that <math>\[\deg f_n \left( x \right) = 0{\rm{ or }}1
 
\]</math>.Then <math>\[f_n \left( x \right) = 1{\rm{ or }}x + 1
 
\]</math>.Then <math>\[f_n \left( x \right) = 1{\rm{ or }}x + 1
−
\]</math>.Check if <math><cmath>f\left( 2 \right) + f\left( 3 \right) = 125</cmath></math> and we can easily get <math><cmath>n = 2</cmath></math> and <math><cmath>f_n \left( x \right) = 1</cmath></math> and <math><cmath>f\left( 5 \right) = \boxed{625}</cmath></math>.
+
\]</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 09:14, 17 March 2008

Here is a completed solution to 2007AIMEII-14. Let $f\left( x \right) = \sum\limits_{i = 0}^n {a_i x^i }$.$\[f\left( 0 \right) = 1 \Rightarrow a_0 = 1 \]$ (Error compiling LaTeX. Unknown error_msg).$f\left( x \right)f\left( {2x^2 } \right) = f\left( {2x^3 + x} \right) \Rightarrow \ldots \Rightarrow a_n = 1$.$f\left( { \pm i} \right)f\left( 2 \right) = f\left( { \mp i} \right) \Rightarrow f\left( { \pm i} \right) = 0 \Rightarrow \left. {\left( {x^2 + 1} \right)} \right|f\left( x \right)$ 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:$\[f_1 \left( x \right) \equiv 1 \]$ (Error compiling LaTeX. Unknown error_msg) or $\left. {\left( {x^2 + 1} \right)} \right|f_1 \left( x \right)$. Let $n$ be an integer and $\[f_n \left( x \right) = \frac{{f\left( x \right)}}{{\left( {x^2 + 1} \right)^n }} \]$ (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 $f\left( 2 \right) + f\left( 3 \right) = 125$ 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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