Difference between revisions of "2007 AMC 12A Problems/Problem 24"

(New page: == Problem == For each integer <math>n>1</math>, let <math>F(n)</math> be the number of solutions to the equation <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. Wha...)
 
(Problem)
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For each integer <math>n>1</math>, let <math>F(n)</math> be the number of solutions to the equation <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>?
 
For each integer <math>n>1</math>, let <math>F(n)</math> be the number of solutions to the equation <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>?
 
 
<math>\mathrm{(A)}\ 2014524  \mathrm{(B)}\ 2015028 \mathrm{(C)}\ 2015033 \mathrm{(D)}\ \2016532 \mathrm{(E)}\ 2017033</math>
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<math>\mathrm{(A)}\ 2014524</math> <math>\mathrm{(B)}\ 2015028</math>  <math>\mathrm{(C)}\ 2015033</math>  <math>\mathrm{(D)}\ 2016532</math>  <math>\mathrm{(E)}\ 2017033</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 20:29, 28 September 2007

Problem

For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?

$\mathrm{(A)}\ 2014524$ $\mathrm{(B)}\ 2015028$ $\mathrm{(C)}\ 2015033$ $\mathrm{(D)}\ 2016532$ $\mathrm{(E)}\ 2017033$

Solution

$F(2)=3$

By looking at various graphs, we obtain that

$F(n+1)=F(n)+1$

$3+4+5+\cdots+2008=2017033$ $\mathrm{(E)}$

See also

2007 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AMC 12 Problems and Solutions
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