▪ Turkey Mathematical Olympiad Team Selection Tests 2013

 Day 1 - 30 March 2013 

1 Let be the number of positive integers less than that are relatively prime to , where is a positive integer. Find all pairs of positive integers such that

2. We put pebbles on some unit squares of a chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each square formed by unit squares contains at least pebbles.

3. Let be the circumcenter and be the incenter of an acute triangle with . Let , , be the midpoints of the sides , , , respectively. Let be the foot of perpendicular from to . Let be the circumcenter of the triangle and be the midpoint of . If , , are collinear, prove that

 Day 2 - 31 March 201 3

1. Find all pairs of integers such that . 

2 Let the incircle of the triangle touch at and be the incenter of the triangle. Let be midpoint of . Let the perpendicular from to meet and at and , respectively. Let the perpendicular from to meet and at and , respectively. Show that 

. 

3. For all real numbers such that 

and , 

determine the least real number satisfying

 Day 3 - 01 April 201 3

1. Let be intersection of the diagonals of convex quadrilateral . It is given that 

. 

If is a point on such that 

, 

show that , , , are concyclic. 

2. Determine all functions such that for all real numbers the following conditions hold:

3. In a country of cities, two way flights between some of the cities are arranged such that it is possible to travel between any two cities by using one or more flights and there are at least flights from each city. Prove that these flights can be distributed between airway companies such that it is possible to travel from any city to another, by using at most one flight of each company.

Πηγή: artofproblemsolving