MathDB

3

Part of 2006 IberoAmerican Olympiad For University Students

Problems(1)

Polynomial recursive sequence - OIMU 2006 Problem 3

Source:

8/30/2010
Let p1(x)=p(x)=4x3−3xp_1(x)=p(x)=4x^3-3x and pn+1(x)=p(pn(x))p_{n+1}(x)=p(p_n(x)) for each positive integer nn. Also, let A(n)A(n) be the set of all the real roots of the equation pn(x)=xp_n(x)=x.
Prove that A(n)⊆A(2n)A(n)\subseteq A(2n) and that the product of the elements of A(n)A(n) is the average of the elements of A(2n)A(2n).
algebrapolynomialinductiontrigonometryalgebra proposed