2007 AMC 12A Problem 24

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

F(n)

counts intersections of

y=\sin x

and

y=\sin nx

on

[0,\pi],

usually two per hump of

\sin nx

F(n)=n+1

except when

n\equiv 1\pmod 4,

where a shared maximum drops it to

n

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

\begin{aligned} &\sum_{n=2}^{2007}F(n) \\ &=\sum_{n=2}^{2007}(n+1) \\ &\quad {}-\#\{n\equiv 1\!\!\pmod 4\}. \end{aligned}

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.

Problem 24 in other years