Posts Tagged ‘function’

IMC 2016 – Day 2 – Problem 8

July 28, 2016 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}.

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IMC 2016 – Day 1 – Problem 1

July 27, 2016 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.

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There isn’t such a function

February 9, 2012 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}.

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Categories: Analysis, Olympiad Tags: ,

Number of even images is very small

June 2, 2011 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 on Subsets

April 29, 2010 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

Reccurent function sequence SEEMOUS 2010

April 26, 2010 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

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Cute Problem with functions

January 21, 2010 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.

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