1987 AMC 12 Problem 24

Question

How many polynomial functions

f

of degree

\ge1

satisfy

f(x^2)=[f(x)]^2=f(f(x))?

0

1

2

finitely many but more than

2

infinitely many

Compare the degrees and leading coefficients of the three polynomials

After proving

f

is a monic quadratic, compare coefficients in

f(x^2)=[f(x)]^2

Let

f

have degree

n\ge1

and leading coefficient

a.

The three expressions have degrees

2n,2n,n^2,

so

n^2=2n

and therefore

n=2.

Comparing leading coefficients in

f(x^2)=[f(x)]^2

gives

a=a^2,

hence

a=1.

Write

f(x)=x^2+bx+c.

Then

x^4+bx^2+c=(x^2+bx+c)^2.

The cubic coefficient forces

b=0,

and then the quadratic coefficient forces

c=0.

Thus the only candidate is

f(x)=x^2,

which indeed satisfies all three expressions. There is exactly one function.

Thus the correct answer is B.

Problem 24 in other years