Home > Olympiad, Problem Solving > Cute Problem with functions

Cute Problem with functions


Let A\subset \mathbb{C} and f:A\rightarrow A A function. Define f_1=f and f_{k+1}=f_k\circ f,\ (\forall)k \in \mathbb{N}^*. Assume that (\exists) \alpha,\ \beta>0 with \alpha+\beta =1 and m,\ n\in \mathbb{N}^* coprimes ( \gcd(m,n)=1), such that \alpha f_m(x)+\beta f_n(x)=x,\ (\forall)x\in A. Find all functions f in the following cases:

i) A=\mathbb{N}.

ii) (\exists)a\in \mathbb{C}^*,\ p\in \mathbb{N}^* such that A=\{z\in \mathbb{C}\ |\ z^p=a\}.

iii) Any set A\subset \mathbb{C} such that there are no three collinear points in A.

Advertisements
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: