## 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

## IMC 2016 – Day 1 – Problem 2

**Problem 2.** Let and be positive integers. A sequence of matrices is *preferred* by Ivan the Confessor if for , but for with . Show that if in al preferred sequences and give an example of a preferred sequence with for each .

## IMC 2016 – Day 1 – Problem 3

**Problem 3.** Let be a positive integer. Also let and be reap numbers such that for . Prove that

## SEEMOUS 2016 – Problems

**Problem 1.** Let be a continuous and decreasing real valued function defined on . Prove that

When do we have equality?

**Problem 2.** a) Prove that for every matrix there exists a matrix such that .

b) Prove that there exists a matrix such that for all .

**Problem 3.** Let be idempotent matrices () in . Prove that

where and is the set of matrices with real entries.

**Problem 4.** Let be an integer and set

Prove that

a)

b) .

Some hints follow.

## IMO 2015 Day 2

**Problem 4.** Triangle has circumcircle and circumcenter . A circle with center intersects the segment at points and , such that , , , and are all different and lie on line in this order. Let and be the points of intersection of and , such that , , , , and lie on in this order. Let be the second point of intersection of the circumcircle of triangle and the segment . Let be the second point of intersection of the circumcircle of triangle and the segment .

Suppose that the lines and are different and intersect at the point . Prove that lies on the line .

**Problem 5.** Let be the set of real numbers. Determine all functions that satisfy the equation

for all real numbers and .

**Problem 6.** The sequence of integers satisfies the conditions:

(i) for all , (ii) for all . Prove that there exist two positive integers and for which

for all integers and such that .

## Vojtech Jarnik Competition 2015 – Problems Category 2

**Problem 1.** Let and be two matrices with real entries. Prove that

provided all the inverses appearing on the left-hand side of the equality exist.

**Problem 2.** Determine all pairs of positive integers satisfying the equation

**Problem 3.** Determine the set of real values for which the following series converges, and find its sum:

**Problem 4.** Find all continuously differentiable functions , such that for every the following relation holds:

where

## Seemous 2015 Problems

**Problem 1.** Prove that for every the following inequality holds:

**Problem 2.** For any positive integer , let the functions be defined by , where . Solve the equation .

**Problem 3.** For an integer , let be matrices satisfying:

where is the identity matrix and is the zero matrix in .

Prove that:

- (a) and .
- (b) and .

**Problem 4.** Let be an open interval which contains and be a function of class such that .

- (i) Prove that there exists such that for .
- (ii) With determined at (i) define the sequence by
Study the convergence of the series for .

**Hints: **