## Balkan Mathematical Olympiad 2018

**Problem 1.** A quadrilateral is inscribed in a circle , where and is not parallel to . Point is the intersection of the diagonals and and the perpendicular from to intersects the segment at the point . If bisects the angle , prove that is a diameter of the circle .

**Problem 2.** Let be a positive rational number. Two ants are initially at the same point in the plane. In the -th minute each of them chooses whether to walk due north, east, south or west and then walks the distance of meters. After a whole number of minutes, they are at the same point in the plane (non necessarily ), but have not taken exactly the same route within that time. Determine all the possible values of .

**Problem 3.** Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, each player chooses a pile with an even number of coins and moves half of the coins of this pile to the other pile. The came ends if a player cannot move, in which case the other player wins.

Determine all pairs of positive integers such that if initially the two piles have and coins, respectively, then Bob has a winning strategy.

**Problem 4.** Find all primes and such that divides .

Source: https://bmo2018.dms.rs/wp-content/uploads/2018/05/BMOproblems2018_English.pdf

## Balkan Mathematical Olympiad 2017 – Problems

**Problem 1.** Find all ordered pairs of positive integers such that:

**Problem 2.** Consider an acute-angled triangle with and let be its circumscribed circle. Let and be the tangents to the circle at points and , respectively, and let be their intersection. The straight line passing through the point and parallel to intersects in point . The straight line passing through the point and parallel to intersects in point . The circumcircle of the triangle intersects in , where is located between and . The circumcircle of the triangle intersects the line (or its extension) in , where is located between and .

Prove that , , and are concurrent.

**Problem 3.** Let denote the set of positive integers. Find all functions such that

for all

**Problem 4.** On a circular table sit students. First, each student has just one candy. At each step, each student chooses one of the following actions:

- (A) Gives a candy to the student sitting on his left or to the student sitting on his right.
- (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions.

(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)

Source: AoPS

## 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.

## Balkan Mathematical Olympiad 2013

**Problem 1.** In a triangle , the excircle opposite to touches at and at and the excircle opposite to touches at and at . Let be the projection of onto , and let be the projection of onto .

Show that the quadrilateral is cyclic.

**Problem 2.** Determine all positive integers such that

**Problem 3.** Let be the set of positive real numbers. Find all functions such that for all positive real numbers and the following three conditions are satisfied:

- (a) ,
- (b) ,
- (c) .

**Problem 4.** In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that when is a friend of , then also is a friend of . We say that different competitors form a *weakly-friendly cycle* if is not a friend of for , and there are no other pairs of non-friends among the components of this cycle.

The following property is satisfied: *for every competitor , and every weakly-friendly cycle of competitors not including , the set of competitors in which are not friends of has at most one element.*

Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same room are friends.

## Balkan Mathematical Olympiad 2011 Problem 4

Let be a convex hexagon of area , whose opposite sides are parallel. The lines , and meet in pairs to determine the vertices of a triangle. Similarly, the lines , and meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least .

BMO 2011 Problem 4

## Balkan Mathematical Olympiad 2011 Problem 3

Let be a finite set of positive integers which has the following property: if is a member of ,then so are all positive divisors of . A non-empty subset of is *good* if whenever and , the ratio is a power of a prime number. A non-empty subset of is *bad* if whenever and , the ratio is not a power of a prime number. A set of an element is considered both *good* and *bad*. Let be the largest possible size of a *good* subset of . Prove that is also the smallest number of pairwise-disjoint *bad* subsets whose union is .

BMO 2011 Problem 3

## Balkan Mathematical Olympiad 2011 Problem 2

Given real numbers such that , show that

When does equality hold?

BMO 2011 Problem 2