## Balkan Mathematical Olympiad – 2016 Problems

**Problem 1.** Find all injective functions such that for every real number and every positive integer ,

**Problem 2.** Let be a cyclic quadrilateral with . The diagonals intersect at the point and lines and intersect at the point . Let and be the orthogonal projections of onto lines and respectively, and let , and be the midpoints of , and respectively. Prove that the second intersection point of the circumcircles of triangles and lies on the segment .

**Problem 3.** Find all monic polynomials with integer coefficients satisfying the following condition: there exists a positive integer such that divides for every prime for which 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 colours so that no rectangle with perimeter contains two squares of the same colour. Show that no rectangle of size or contains two squares of the same colour.

## Sierpinski’s Theorem for Additive Functions – Simplification

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

2. For every solution of the Cauchy functional equation there exist two non-trivial solutions of the same equation, such that and have the Darboux property and .

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

**Problem 7.** Suppose that is an additive function (that is for all ) for which is bounded of some nonempty subinterval of . Prove that is continuous.

## Miklos Schweitzer 2013 Problem 8

**Problem 8.** Let be a continuous and strictly increasing function for which

for all ( denotes the inverse of ). Prove that there exist real constants and such that for all .

## IMO 1996 Day 1

**Problem 1** We are given a positive integer and a rectangular board with dimensions . The rectangle is divided into a grid of 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 . The task is to find a sequence of moves leading from the square with as a vertex to the square with as a vertex.

(a) Show that the task cannot be done if is divisible by or .

(b) Prove that the task is possible when .

(c) Can the task be done when ?

## Balkan Mathematical Olympiad 2012 Problem 4

Let be the set of positive integers. Find all functions such that the following conditions both hold:

(i) for every positive integer ,

(ii) divides whenever and are different positive integers.

*Balkan Mathematical Olympiad 2012 Problem 4*