2007 AMC 12A Problems/Problem 24

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, for most of the graphs

$F(n) = n + 1$

However, when $n \equiv 1 \pmod{4}$ , the middle apex of the sine curve touches the sine curve at the top only one time (instead of two), so we get here $F(n) = n$ .

$3+4+5+5+7+8+9+9+\cdots+2008$ $= (1+2+3+4+5+\cdots+2008) - 3 - 501$ $= \frac{(2008)(2009)}{2} - 504 = 2016532$ $\mathrm{(D)}$

2007 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2007 AMC 12A Problems/Problem 24

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