Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials \(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\). Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\). Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\).
Problem
Source: 38th Brazilian MO 2016 - Second Day, Problem 5
Tags: Brazilian Math Olympiad 2016, algebra, polynomial
tk1
23.11.2016 19:52
See here.
LittleGlequius
19.01.2017 15:37
My Solution: Let $Q(x)=\frac{P(x)}{2016}$. Then $P_{1}(x)=Q(x)$ , $P_{n+1}(x)=Q(P_{n}(x))$. Define $P_{0}(x)=x$. See that $Q(x)=\frac{(2x+3)^{2}-3024}{2016}=\frac{(x+\frac{3}{2})^{2}}{504}-\frac{3}{2}$. $\frac{Q(x)+\frac{3}{2}}{504}=(\frac{x+\frac{3}{2}}{504})^{2}$ Then, for every positive integer $n$, $\frac{P_{n}(x)+\frac{3}{2}}{504}=(\frac{P_{n-1}(x)+\frac{3}{2}}{504})^{2}= ...=(\frac{P_{0}(x)+\frac{3}{2}}{504})^{2^{n}}$. Then, $P_{n}(x)=504(\frac{x+\frac{3}{2}}{504})^{2^{n}}-\frac{3}{2}$ (a)$P_{n}(-\frac{3}{2})=-\frac{3}{2}<0$. Take $r=-\frac{3}{2}$ (b)$P_{n}(m)<0$ iff $(\frac{m+\frac{3}{2}}{504})^{2^{n}}<\frac{1}{336}$. This holds for infinitely positive integers $n$ iff $|\frac{m+\frac{3}{2}}{504}|<1$ iff $-505.5<m<502.5$ . There are $1008$ integers with this property.
npmt
20.12.2021 07:48
LittleGlequius wrote: My Solution: Let $Q(x)=\frac{P(x)}{2016}$. Then $P_{1}(x)=Q(x)$ , $P_{n+1}(x)=Q(P_{n}(x))$. Define $P_{0}(x)=x$. See that $Q(x)=\frac{(2x+3)^{2}-3024}{2016}=\frac{(x+\frac{3}{2})^{2}}{504}-\frac{3}{2}$. $\frac{Q(x)+\frac{3}{2}}{504}=(\frac{x+\frac{3}{2}}{504})^{2}$ Then, for every positive integer $n$, $\frac{P_{n}(x)+\frac{3}{2}}{504}=(\frac{P_{n-1}(x)+\frac{3}{2}}{504})^{2}= ...=(\frac{P_{0}(x)+\frac{3}{2}}{504})^{2^{n}}$. Then, $P_{n}(x)=504(\frac{x+\frac{3}{2}}{504})^{2^{n}}-\frac{3}{2}$ (a)$P_{n}(-\frac{3}{2})=-\frac{3}{2}<0$. Take $r=-\frac{3}{2}$ (b)$P_{n}(m)<0$ iff $(\frac{m+\frac{3}{2}}{504})^{2^{n}}<\frac{1}{336}$. This holds for infinitely positive integers $n$ iff $|\frac{m+\frac{3}{2}}{504}|<1$ iff $-505.5<m<502.5$ . There are $1008$ integers with this property. I don't get this part $|\frac{m+\frac{3}{2}}{504}|<1$ . Can you please explain me more about it?