# 19th Turkish Mathematical Olympiad

## Day 1, Decmeber 3, 2011

** Problem 1 **

Let \( n\geq 2 \) be an integer and \( E=\{1,2,\dots, n\} \). If \( A_1 \), \( \dots \), \( A_k \) are subsets of \( E \) and exactly one of
\( A_i\cap A_j \), \( A_i^C\cap A_j \), \( A_i\cap A_j^C \), and \( A_i^C\cap A_j \) is empty for all \( 1\leq i< j\leq k \), then determine the maximum possible value of \( k \).

*Remark.* For \( A\subseteq E \), \( A^C \) denotes the elements of \( E \) which are not included in \( A \).

** Problem 2 **

Let \( D \) be a point on the side \( BC \) of \( \triangle ABC \) different from the vertices and let \( E \) be the midpoint of \( CD \). The line perpendicular to \( BC \) at \( E \) intersects the side \( AC \) at point the \( F \) satisfying \( AF\cdot BC=AC\cdot EC \). Let \( G \) be the second point where the circumcircle of the triangle \( ADC \) meets the side \( AB \). Prove that the tangent line of the circumcircle of the triangle \( AGF \) at \( F \) is also tangent to the circumcircle of the triangle \( BGE \).

** Problem 3 **

Prove that the inequality \[ \frac1{x+y^{20}+z^{11}}+\frac1{y+z^{20}+x^{11}}+\frac1{z+x^{20}+y^{11}}\leq 11\] holds for all positive real numbers \( x \), \( y \), and \( z \) that satisfy \( xyz=1 \).

## Day 2, Decmeber 4, 2011

** Problem 4 **

Let \( a_{n+1}=a_n^3-2a_n^2+2 \) for all \( n\geq 1 \) and \( a_1=5 \). Prove that if \( p\equiv 3 \) (mod \( 4 \)) is a prime divisor of \( a_{2011}+1 \), then \( p=3 \).

** Problem 5 **

Given two regular polygonal regions in the plane \( M \) and \( N \), let \( K(M,N) \) be the collection of the midpoints of the line segments where one endpoint belongs to \( M \) and the other belongs to \( N \). Determine all pairs \( (M,N) \) for which \( K(M,N) \) is a regular convex polygonal region.

** Problem 6 **

Between any two cities of country \( A \) consisting of 2011 cities and country \( B \) consisting of 2011 cities there is a unique direct two way flight organized by some airway company. For each given city there are at most 19 different airway companies operating flights from this city. Determine the maximum possible value of the integer \( k \) such that no matter how these flights are arranged there are \( k \) cities connected (not necessarily directly) only by the flights of some fixed airway company.

## Acknowledgments

Material received from Boš Beleš.