### Archive

Posts Tagged ‘function’

## IMC 2016 – Day 2 – Problem 8

Problem 8. Let ${n}$ be a positive integer and denote by ${\Bbb{Z}_n}$ the ring of integers modulo ${n}$. Suppose that there exists a function ${f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n}$ satisfying the following three properties:

• (i) ${f(x) \neq x}$,
• (ii) ${x = f(f(x))}$,
• (iii) ${f(f(f(x+1)+1)+1) = x}$ for all ${x \in \Bbb{Z}_n}$.

Prove that ${n \equiv 2}$ modulo ${4}$.

## IMC 2016 – Day 1 – Problem 1

Problem 1. Let ${f:[a,b] \rightarrow \Bbb{R}}$ be continuous on ${[a,b]}$ and differentiable on ${(a,b)}$. Suppose that ${f}$ has infinitely many zeros, but there is no ${x \in (a,b)}$ with ${f(x)=f'(x) = 0}$.

• (a) Prove that ${f(a)f(b)=0}$.
• (b) Give an example of such a function.

## There isn’t such a function

Prove that there is no continuous function $f: \Bbb{R} \to \Bbb{R}$ such that $f(\Bbb{Q}) \subset \Bbb{R}\setminus \Bbb{Q}$ and $f(\Bbb{R}\setminus \Bbb{Q}) \subset \Bbb{Q}$.

## Number of even images is very small

I found on Math Stack Exchange a generalisation of a problem I proposed in the Romanian Mathematical Olympiad:

Let $f:[0,1]\to [0,1]$  be a continuous function such that $f(0)=0$ and $f(1)=1$. Moreover assume $f^{-1}(\{x\})$ is finite for all $x$. Prove that $E:=\{x\in [0,1]: |f^{-1}(\{x\})|\,\mbox{ is even} \}$ is countable.

My problem was to prove that there doesn’t exist a function $f:\Bbb{R} \to \Bbb{R}$ with the property that the equation $f(x)=y$ has an even number of solutions for all $y \in \Bbb{R}$.

Categories: Analysis Tags:

## Function on Subsets

Let $f:2^S \to \mathbb{R}$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=f(S\setminus A)$ and $\max \{f(A),f(B)\} \geq f(A\cup B)$ for all subsets $A,B \subset S$, then $f$ assumes at most $|S|$ distinct values.
Miklos Schweitzer, 2001

## Reccurent function sequence SEEMOUS 2010

Suppose $f_0:[0,1]\to \mathbb{R}$ is a continuous function, and define the sequence $(f_n)_n, f_n:[0,1]\to \mathbb{R}$ in the following way:

$f_n(x)=\int_0^x f_{n-1}(t)dt,\ \forall x \in [o,1]$.

a) Prove that the series $\sum_{n\geq 0}f_n(x)$ converges for any $x \in [0,1]$.

b) Find an explicit formula in terms of $x$ for the above series.

Seemous 2010, Problem 1

## 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 $A$.