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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<CD}. 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.

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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

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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

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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}?

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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

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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,...\}.

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