### Archive

Posts Tagged ‘functional equation’

## Balkan Mathematical Olympiad – 2016 Problems

Problem 1. Find all injective functions ${f: \mathbb R \rightarrow \mathbb R}$ such that for every real number ${x}$ and every positive integer ${n}$,

$\displaystyle \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$

Problem 2. Let ${ABCD}$ be a cyclic quadrilateral with ${AB. The diagonals intersect at the point ${F}$ and lines ${AD}$ and ${BC}$ intersect at the point ${E}$. Let ${K}$ and ${L}$ be the orthogonal projections of ${F}$ onto lines ${AD}$ and ${BC}$ respectively, and let ${M}$, ${S}$ and ${T}$ be the midpoints of ${EF}$, ${CF}$ and ${DF}$ respectively. Prove that the second intersection point of the circumcircles of triangles ${MKT}$ and ${MLS}$ lies on the segment ${CD}$.

Problem 3. Find all monic polynomials ${f}$ with integer coefficients satisfying the following condition: there exists a positive integer ${N}$ such that ${p}$ divides ${2(f(p)!)+1}$ for every prime ${p>N}$ for which ${f(p)}$ is a positive integer.

Problem 4. The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of ${1201}$ colours so that no rectangle with perimeter ${100}$ contains two squares of the same colour. Show that no rectangle of size ${1\times1201}$ or ${1201\times1}$ contains two squares of the same colour.

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## Sierpinski’s Theorem for Additive Functions – Simplification

February 3, 2014 Leave a comment

1. If ${f}$ is a solution of the Cauchy functional equation which is surjective, but not injective, then ${f}$ has the Darboux property.

2. For every solution ${f}$ of the Cauchy functional equation there exist two non-trivial solutions ${f_1,f_2}$ of the same equation, such that ${f_1}$ and ${f_2}$ have the Darboux property and ${f=f_1+f_2}$.

These two results were proven in this post. The version presented here is a simplified one, identifying exactly what we need in order to obtain the desired results.

## Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.

Miklos Schweitzer 2013

Categories: Algebra, Analysis, Geometry

## Miklos Schweitzer 2013 Problem 8

November 9, 2013 1 comment

Problem 8. Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which

$\displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$

for all ${x,y \in \Bbb{R}}$ (${f^{-1}}$ denotes the inverse of ${f}$). Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}$.

Miklos Schweitzer 2013

## IMO 1996 Day 1

June 9, 2012 1 comment

Problem 1 We are given a positive integer ${ r}$ and a rectangular board ${ ABCD}$ with dimensions ${ AB = 20, BC = 12}$. The rectangle is divided into a grid of ${ 20 \times 12}$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is ${ \sqrt {r}}$. The task is to find a sequence of moves leading from the square with ${ A}$ as a vertex to the square with ${ B}$ as a vertex.

(a) Show that the task cannot be done if ${ r}$ is divisible by $2$ or $3$.

(b) Prove that the task is possible when ${ r = 73}$.

(c) Can the task be done when ${ r = 97}$?

## Balkan Mathematical Olympiad 2012 Problem 4

April 29, 2012 4 comments

Let ${\mathbb{Z}^+}$ be the set of positive integers. Find all functions ${f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+}$ such that the following conditions both hold:

(i) ${f(n!)=f(n)!}$ for every positive integer ${n}$,

(ii) ${m-n}$ divides ${f(m)-f(n)}$ whenever ${m}$ and ${n}$ are different positive integers.

Balkan Mathematical Olympiad 2012 Problem 4

Categories: Olympiad Tags:

## Functional equation on positive integers

February 9, 2012 4 comments

Let $p$ be a fixed positive integer. Find all functions $f : \Bbb{N} \to \Bbb{N}$ such that $\forall n \in \Bbb{N}$ we have $f(n+1) > \underbrace{f\circ f \circ...\circ f}_{p \text{ times}}(n)$.

$\Bbb{N}=\{0,1,2,3,...\}$.