1985 AMC 12 Problem 21

Attempt Problem 21 of the 1985 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 1985 AMC 12 solutions, or check the answer key.

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

21.

How many integers

x

satisfy the equation

(x^2-x-1)^{x+2}=1?

2

3

4

5

none of these

An integer power can equal

1

when the base is

1,

when the base is

-1

with even exponent, or when a nonzero base has exponent

0

Solve those three cases separately and check every candidate

If the base is

1,

then

x^2-x-1=1,

giving

x=-1

and

x=2.

If the base is

-1,

then

x(x-1)=0;

only

x=0

makes the exponent

x+2

even. If the exponent is

0,

then

x=-2,

and its base is

5,

so it also works. Thus the four solutions are

-2,

-1,

0,

and

2.

Therefore the correct answer is C.

Problem 21 in other years