1982 AMC 12 Problem 12

Attempt Problem 12 of the 1982 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 1982 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

Let

f(x)=ax^7+bx^3+cx-5,

where

a,

b,

and

c

are constants. If

f(-7)=7,

then

f(7)

equals

-17

-7

14

21

not uniquely determined

Separate the constant term from the odd-powered terms

g(x)=f(x)+5,

compare

g(7)

and

g(-7)

Let

g(x)=ax^7+bx^3+cx,

which is odd. Since

f(-7)=g(-7)-5=7,

g(-7)=12.

Therefore

g(7)=-12,

and

f(7)=-12-5=-17.

Therefore, the correct answer is A.

Problem 12 in other years