# 2007 AMC 12A Problems/Problem 24

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## 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 $\sum_{n=2}^{2007} F(n)$?

$\mathrm{(A)}\ 2014524 \mathrm{(B)}\ 2015028 \mathrm{(C)}\ 2015033 \mathrm{(D)}\ \2016532 \mathrm{(E)}\ 2017033$ (Error compiling LaTeX. ! Undefined control sequence.)

## 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 (Problems • Answer Key • Resources) Preceded byProblem 23 Followed byProblem 25 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions
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