MATH PROBLEMS OF 2002

 January 2002 Question : Find the minimum of   if   and  . Solution 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. Solution 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. Solution Congratulations Baatar Tsolman Middle East Technical University, Ankara

 June 2002 Question : Is there an integer n such that  is a rational number? Solution 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

 July-August 2002 Question : Non-negative real numbers a, b, and c satisfy  . Prove that   Solution 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 x1, x2, … , x2002  be some points lying on a unit circle and dij be the distance between xi and xj . Let  be the sum taken over all possible pairs (xi ,xj)              for    i, j = 1, 2, … , 2002 and   i < j .  Find the maximum of S over all possible distributions of x1, x2, … , x2002   Solution 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 : Solution 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 2n, where n is a natural number. Prove that  A ≠ 2k for all k > n . Solution 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 = p1.p2. … pr , where pi is prime for each i = 1, 2, … , r , define f (n) = 1 + p1 + p2 + … + pr .   Prove that for any natural number k, the sequence a1 = k, am = f(am-1) , m = 2, 3, … is periodic.   Solution 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