Problem

Source: Iranian National Olympiad (3rd Round) 2003

Tags: algebra, polynomial, induction, algebra proposed



Let cC and Ac={pC[z]|p(z2+c)=p(z)2+c}. a) Prove that for each cC, Ac is infinite. b) Prove that if pA1, and p(z0)=0, then |z0|<1.7. c) Prove that each element of Ac is odd or even. Let fc=z2+cC[z]. We see easily that Bc:={z,fc(z),fc(fc(z)),} is a subset of Ac. Prove that in the following cases Ac=Bc. d) |c|>2. e) cQZ. f) c is a non-algebraic number g) c is a real number and c[2,14].