MATH PROBLEMS OF 2002
January 2002 
Question : Find the minimum of if and . 

Congratulations  
Serhat Doğan  Özel Şehzade Mehmet Lisesi, Manisa  
Murat Ak  Bilkent Universitesi, Ankara  
Yankı Lekili  Bilkent Universitesi, Ankara  
Erkan Özkan  Özel Nilüfer Lisesi, Bursa 
April 2002 
Question : Prove that the equation
has no solution in natural numbers.


Congratulations  
Suat Gumussoy  The Ohio State University  
Murat Ak  Bilkent University, Ankara  
Baatar Tsolman  Middle East Technical University, Ankara  
Koksal Dinc  Hacettepe University  
Erkan Ozkan  Ozel Nilufer Lisesi, Bursa  
Oztekin Bakir  Bartin 
May 2002 
Question :
There is a finite number of towns in a country. They are connected by one direction roads. It is known that for any two towns, one of them can be reached from the other one. Prove that there is a town such that all the remaining towns can be reached from it.


Congratulations 

Baatar Tsolman  Middle East Technical University, Ankara  
June 2002 
Question : Is there an integer n such that is a rational number?


Congratulations


Ali Yildiz  Bilkent University, Ankara  
Vivek Kumar Mehra  Mumbai, India  
Serhat Dogan  Ozel Sehzade Mehmet Lisesi, Manisa  
Ha Duy Hung  Hanoi University of Education, Vietnam  
Sener Ozturk  
Oztekin Bakir  Bartin  
Hakan Ozaydin  Middle East Technical University, Ankara  
Aylin Tokuc  Bilkent University, Ankara  
Murat Ak  Bilkent University, Ankara 
JulyAugust 2002 
Question : Nonnegative real numbers a, b, and c satisfy . Prove that


Congratulations  
Vejdi Hasanov  Sumen University, Bulgaria  
Ha Duy Hung  Hanoi University of Education, Vietnam  
Murat Ak  Bilkent University, Ankara  
Vivek Kumar Mehra  Mumbai, India  
Ali Yıldız  Bilkent University, Ankara  
Jacob Tsimerman  Toronto, Canada  
Şener Öztürk  İstanbul  
Beata Stehlikova  Comenius University, Bratislava, Slovakia 
September 2002 
Question :
Let x_{1}, x_{2}, … , x_{2002 }be some points lying on a unit circle and d_{ij} be the distance between x_{i} and x_{j} . Let be the sum taken over all possible pairs (x_{i} ,x_{j})
for i, j = 1, 2, … , 2002 and i < j . Find the maximum of S over all possible distributions of x_{1}, x_{2}, … , x_{2002 }


Congratulations  

Ha Duy Hung 
Hanoi University of Education, Vietnam 

Murat Ak  Bilkent University, Ankara  
K. Zhereb 
Moscow Institute of Physics and Technology, Moscow 

Mustafa Turgut  Isparta  
October 2002 
Question : Find all real solutions of the following equation :


Congratulations  
Hung Ha Duy 
Hanoi University of Education, Vietnam 

Mustafa Turgut 
Isparta 

Vejdi Hasanov 
Shumen University, Bulgaria 

Beata Stehlikova  Comenius University, Bratislava, Slovakia  
Jacob Tsimerman  Toronto, Canada  
Murat Ak  Bilkent University, Ankara  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 



Eaturu Sribar 
Indian Institute of Technology, Mumbai, India 

Ignas Buvitis 
Lithuania 

Athanasios Papaioannou 
Thessaloniki, Greece 

Şener Öztürk  İstanbul  
Birol Bakay 
Bilkent University, Ankara 
November 2002 
Question : Let A be a number obtained by some rearrangement of the digits of 2^{n}, where n is a natural number. Prove that A ≠ 2^{k} for all k > n . 

Congratulations  
Jacob Tsimerman  Toronto, Canada  
Beata Stehlikova  Comenius University, Bratislava, Slovakia  
Birol Bakay  Bilkent University, Ankara  
Mustafa Öztekin  Boğaziçi University, Istanbul  
Umut Işık  Bilkent University, Ankara  
Vivek Kumar Mehra  Mumbai, India  
Ali Yıldız  Bilkent University, Ankara  
Yiğit Subaşı  Bilkent University, Ankara  
Erdem Özcan  Bilkent University, Ankara 
December 2002 
Question : For each natural number n = p_{1}.p_{2}. … p_{r} , where p_{i} is prime for each i = 1, 2, … , r , define f (n) = 1 + p_{1} + p_{2} + … + p_{r} .
Prove that for any natural number k, the sequence a_{1} = k, a_{m} = f(a_{m1}) , m = 2, 3, … is periodic.


Congratulations  
Vivek Kumar Mehra  Mumbai, India  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Athanasios Papaioannou 
Thessaloniki, Greece 

Julien Santini  Universite de Provence, France  
Emre Çakır  Bilkent University, Ankara  
Erdem Özcan  Bilkent University, Ankara  
Umut Işık  Bilkent University, Ankara  
David Anderson  Middle East Technical University, Ankara  
Mustafa Öztekin  Boğaziçi University, Istanbul  
Jacob Tsimerman  Toronto, Canada  
Beata Stehlikova  Comenius University, Bratislava, Slovakia 