Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

Problem A. 735. (November 2018)

A. 735. For any function $\displaystyle f\colon [0,1]\to[0,1]$, let $\displaystyle P_n(f)$ denote the number of fixed points of the function

$\displaystyle \underbrace{f\big(\ldots f}_n(x)\ldots\big),$

i.e., the number of points $\displaystyle x\in[0,1]$ satisfying $\displaystyle \underbrace{f\big(\ldots f}_{n}(x)\ldots\big)=x$. Construct a piecewise linear, continuous, surjective function $\displaystyle f\colon [0,1]\to[0, 1]$ such that for a suitable number $\displaystyle {2<A<3}$, the sequence $\displaystyle \frac{P_n(f)}{A^n}$ converges.

Based on the 8th problem of the Miklós Schweitzer competition, 2018

(7 pont)

Deadline expired on December 10, 2018.

Statistics:

 4 students sent a solution. 7 points: Schrettner Jakab. 2 points: 3 students.

Problems in Mathematics of KöMaL, November 2018