Question

Consider those functions $f$ that satisfy $f(x+4)+f(x-4)=f(x)$ for all real $x.$ Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p.$

$8$

$12$

$16$

$24$

$32$

Answer: D

Concepts:functional equations recurrences periodicity

Difficulty rating: 2180

Small Hint:

For fixed $x$ , study the sequence $u_n=f(x+4n)$

Big Hint:

Use $u_{n+1}+u_{n-1}=u_n$ repeatedly to find when every initial pair returns

Solution:

For $u_n=f(x+4n),$ the equation is $u_{n+1}=u_n-u_{n-1}.$ Starting from $u_0,u_1,$ the sequence is $\begin{gathered} u_0,u_1,u_1-u_0,-u_0,\\ -u_1,u_0-u_1,u_0,u_1,\ldots, \end{gathered}$ so every such function has period $6\cdot4=24.$ This is least because $f(x)=\sin(\frac{\pi x}{12})$ satisfies the equation and has fundamental period $24.$ Hence D is correct.

1997 AMC 12 Problem 27

Answer: D

Difficulty rating: 2180

Small Hint:

Big Hint:

Solution:

Problem 27 in Other Years