1997 AMC 12 第 27 题

Question

考虑所有对任意实数

x

都满足

f(x+4)+f(x-4)=f(x)

的函数

f

。每个这样的函数都是周期函数，并且存在一个它们共有的最小正周期

p

。求

p

。

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

固定

x

，研究数列

u_n=f(x+4n)

For fixed

x

, study the sequence

u_n=f(x+4n)

反复使用

u_{n+1}+u_{n-1}=u_n

，找出任意初始二元组何时恢复原状

Use

u_{n+1}+u_{n-1}=u_n

repeatedly to find when every initial pair returns

对

u_n=f(x+4n)

，原方程化为

u_{n+1}=u_n-u_{n-1}

。从

u_0,u_1

出发，数列依次为

\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}\text{，}

因此每个这样的函数都有周期

6\cdot4=24

。又因为

f(x)=\sin(\frac{\pi x}{12})

满足原方程且最小正周期为

24

，所以这个公周期不能再小。故正确答案是 D。

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.

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