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IMC 2016 – Day 2 – Problem 8

July 28, 2016 beni22sof 2 comments

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 ${4}$ .

Categories: Algebra, Group Theory, Olympiad, Uncategorized Tags: function, groups, IMC, permutation

IMC 2016 – Day 1 – Problem 1

July 27, 2016 beni22sof Leave a comment

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.

Categories: Analysis, Olympiad Tags: Analysis, convergence, derivative, function

There isn’t such a function

February 9, 2012 beni22sof 2 comments

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 $f(\Bbb{R}\setminus \Bbb{Q}) \subset \Bbb{Q}$ .

Categories: Analysis, Olympiad Tags: continuous, function

Number of even images is very small

June 2, 2011 beni22sof Leave a comment

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

Function on Subsets

April 29, 2010 beni22sof Leave a comment

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

Categories: Combinatorics, Olympiad, Problem Solving Tags: function, miklos schweitzer

Reccurent function sequence SEEMOUS 2010

April 26, 2010 beni22sof Leave a comment

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

Categories: Analysis, Olympiad, Uncategorized Tags: continuous, function, SEEMOUS

Cute Problem with functions

January 21, 2010 beni22sof Leave a comment

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 $A$ .

Categories: Olympiad, Problem Solving Tags: function, olympiad

Measurable function again

January 7, 2010 beni22sof 2 comments

Suppose $f:[a,b]\to \mathbb{R}$ is a measurable function. Prove that there exists a sequence of polynomials $(P_n)$ such that $P_n \to f$ almost uniformly, which means that for any $\varepsilon >0$ , we can find a set $A_\varepsilon$ with measure smaller than $\varepsilon$ such that $P_n \to f$ uniformly on $[a,b]\setminus A_\varepsilon$ .
Hint: Use Lusin’s theorem and Weierstrass’ approximation theorem, for continuous functions.

Categories: Analysis, Measure Theory, Undergraduate Tags: approximation, function, measure, uniformly

Lusin’s Theorem

January 6, 2010 beni22sof Leave a comment

Suppose $f$ is measurable and finite valued on $E$ , with $E$ of finite measure. Then for every $\varepsilon >0$ there exists a closed set $F_\varepsilon \subset E$ , with $m(E\setminus F_\varepsilon) \leq \varepsilon$ and such that $f | _{F_\varepsilon}$ is continuous.

This theorem states that any measurable function in a space with finite measure can be made almost continuous. Littlewood said that Every measurable function is nearly continuous.

Categories: Analysis, Measure Theory, Undergraduate Tags: function, measure

Egorov’s Theorem

January 6, 2010 beni22sof Leave a comment

Suppose $(f_k)$ is a sequence of measurable functions defined on a measurable set $E$ with $m(E)<\infty$ , we can find a closed set $A_\varepsilon \subset E$ such that $m(E\setminus A_\varepsilon)\leq \varepsilon$ and $f_k \to f$ uniformly on $E\setminus A_\varepsilon$ .

This theorem states that in spaces with finite measure, almost everywhere convergence is equivalent to almost uniformly convergence.

Categories: Analysis, Measure Theory, Undergraduate Tags: function, measure, uniformly

Jensen convexity

October 26, 2009 beni22sof Leave a comment

A function defined on an interval $I \subset \mathbb{R}$ (with non-void interior) is called $\lambda$ -convex for a $\lambda \in (0,1)$ if $f(\lambda x + (1-\lambda) y) \leq \lambda f(x)+(1-\lambda)f(y),\ \forall x,y \in I$ .

The function $f$ defined on an interval is called convex if it’s $\lambda$ -convex for all $\lambda \in (0,1)$ . A function is called Jensen convex if it is $\frac{1}{2}$ -convex.

Prove that any $\lambda$ -convex continuous function is $\frac{1}{2}$ convex, any convex function is continuous on the interior of its definition interval. Furthermore, if $f$ is Jensen convex and continuous, then $f$ is convex. Therefore, if $f$ is continuous and $\lambda$ convex then $f$ is convex.

Prove that there exist $\lambda$ -convex functions which are not convex.

Categories: Analysis, Problem Solving Tags: continuous, convex, function, Jensen

Interesting property for a differentiable function

October 1, 2009 beni22sof Leave a comment

Prove that if $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function on $\mathbb{R}$ , then the set of continuity points of $f^\prime$ is dense in $\mathbb{R}$ .

Categories: Analysis Tags: continuity, dense, differentiable, function

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IMC 2016 – Day 2 – Problem 8

IMC 2016 – Day 1 – Problem 1

There isn’t such a function

Number of even images is very small

Function on Subsets

Reccurent function sequence SEEMOUS 2010

Cute Problem with functions

Measurable function again

Lusin’s Theorem

Egorov’s Theorem

Jensen convexity

Interesting property for a differentiable function

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