Question

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

$2{,}014{,}524$

$2{,}015{,}028$

$2{,}015{,}033$

$2{,}016{,}532$

$2{,}017{,}033$

Solution:

On each interval where $\sin nx\ge 0,$ the graphs of $\sin x$ and $\sin nx$ meet twice, unless they share the value $1$ there, in which case they meet once. Counting the humps and the endpoint at $(\pi,0)$ gives

$F(n)=n+1$ when $n$ is even or $n\equiv 3\pmod 4,$ and $F(n)=n$ when $n\equiv 1\pmod 4.$

Thus $\sum_{n=2}^{2007}F(n)=\sum_{n=2}^{2007}(n+1)-\#\{n\equiv 1\!\!\pmod 4\}.$ The first sum is $2{,}017{,}033,$ and there are $501$ values $n\equiv 1\pmod 4$ in the range, giving $2{,}017{,}033-501=2{,}016{,}532.$

Thus, the correct answer is D.

2007 AMC 12A Problem 24

Solution:

Problem 24 in Other Years