1988 AMC 12 Problem 30

Question

Let

f(x)=4x-x^2.

Given

x_0,

consider the sequence defined by

x_n=f(x_{n-1})

for all

n\ge1.

For how many real numbers

x_0

will the sequence

x_0,

x_1,

x_2,

\ldots

take on only a finite number of different values?

0

1

or

2

3,

4,

5

or

6

more than

6

but finitely many

infinitely many

Answer: E

Concepts:function iteration preimages induction finite orbits

Difficulty rating: 2930

Small hint:

Start with

0,

then find numbers mapping successively to

0

Big hint:

For every

a\le4,

solve

4x-x^2=a

and choose a new real preimage

Solution:

The starting values

0,4,2

give finite orbits

0

,

4\to0

, and

2\to4\to0.

More generally, if

a_n

begins a finite chain ending at

0,

solve

4a_{n+1}-a_{n+1}^2=a_n.

Its real solutions are

a_{n+1}=2\pm\sqrt{4-a_n}.

Choosing a preimage not already in the chain extends it by one new value. Repeating produces infinitely many distinct starting values with finite orbits.

Thus the correct answer is E.