MathDB

Polynomial recursive sequence - OIMU 2006 Problem 3

Source:

August 30, 2010

algebrapolynomialinductiontrigonometryalgebra proposed

Problem Statement

Let

p_1(x)=p(x)=4x^3-3x

and

p_{n+1}(x)=p(p_n(x))

for each positive integer

n

. Also, let

A(n)

be the set of all the real roots of the equation

p_n(x)=x

.
Prove that

A(n)\subseteq A(2n)

and that the product of the elements of

A(n)

is the average of the elements of

A(2n)