1977 AMC 12 Problem 22

Attempt Problem 22 of the 1977 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1977 AMC 12 solutions, or check the answer key.

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22.

f(x)

is a real-valued function of the real variable

x,

and

f(x)

is not identically zero, and for all

a

and

b

\begin{gathered} f(a+b)+f(a-b)\\ =2f(a)+2f(b), \end{gathered}

then for all

x

and

y

f(0)=1

f(-x)=-f(x)

f(-x)=f(x)

f(x+y)=f(x)+f(y)

there is a positive number

T

such that

f(x+T)=f(x)

First substitute

a=b=0

Next set

a=0

and

b=x

Setting

a=b=0

gives

2f(0)=4f(0),

f(0)=0.

Now set

a=0

and

b=x.

The equation becomes

\begin{aligned} f(x)+f(-x) &=2f(0)+2f(x)\\ &=2f(x), \end{aligned}

and hence

f(-x)=f(x)

for every real

x.

Therefore, the correct answer is C.

Problem 22 in other years