For each [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n>1$, let $F(n)$ be the number of solutions to the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?
$\mathrm{(A)}\ 2014524\qquad\mathrm{(B)}\ 2015028\qquad\mathrm{(C)}\ 2015033\qquad\mathrm{(D)}\ 2016532\qquad\mathrm{(E)}\ 2017033$
$F(2)=3$
By looking at various graphs, we obtain that, for most of the graphs
$F(n) = n + 1$
Notice that the solutions are basically reflections across $x = \frac{\pi}{2}$.
However, when $n \equiv 1 \pmod{4}$, the middle apex of the [sine](https://artofproblemsolving.com/wiki/index.php/Sine) curve touches the sine curve at the top only one time (instead of two reflected points), so we get here $F(n) = n$.
$3+4+5+5+7+8+9+9+\cdots+2008$
$= (1+2+3+4+5+\cdots+2008) - 3 - 501$
$= \frac{(2008)(2009)}{2} - 504 = 2016532$ $\mathrm{(D)}$