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

$F(n+1)=F(n)+1$

$3+4+5+\cdots+2008=2017033$ $\mathrm{(E)}$

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

Page

Toolbox

Search

2007 AMC 12A Problems/Problem 24

Problem

Solution

See also