MATH PROBLEMS OF 2003

 January 2003 Question : Let a1, a2, … , an be n natural numbers satisfying ai ≤ i for each i = 1, … ,n . If  is even, prove that at least one  expression of the form a1 ± a2 ± a3 ± … ± an is equal to zero. Solution Congratulations Julien Santini Universite de Provence, France Athanasios Papaioannou Thessaloniki, Greece Stojan Trajanovski High School "RJ Korcagin" Skopje, R. Macedonia Jacob Tsimerman Toronto, Canada Hung Ha Duy Hanoi University of Education, Vietnam Ti Yin Toronto, Canada

 February 2003 Question : Prove that the equation  has infinitely many natural solutions. Solution Congratulations Hung Ha Duy Hanoi University of Education, Vietnam Umut Işık Bilkent University, Ankara Jacob Tsimerman Toronto, Canada Stojan Trajanovski High School "RJ Korcagin" Skopje, R. Macedonia Athanasios Papaioannou Thessaloniki, Greece Beata Stehlikova Comenius University, Bratislava, Slovakia Janos Kramar Toronto, Canada Erdem Özcan Bilkent University, Ankara Erkan Melih Şensoy TED Zonguldak College Julien Santini Universite de Provence, France Ahmet Şensoy Bilkent University, Ankara Vejdi Hasanov Shumen University, Bulgaria Mustafa Öztekin Bosporus University, Istanbul Richard Pinch Cheltenham, United Kingdom Öztekin Bakır Middle East Technical University, Ankara Eaturu Sribar Indian Institute of Technology, Mumbai, India Eren Çetin Hacettepe University, Ankara Ali Yıldız Bilkent University, Ankara

 March 2003 Question : Consider a collection of 26 stones weighted 1, 2, 3, … , 26 grams. Let us denote it by . Prove that a)  There is a subset  of , consisting of 6 stones, which does not contain any two separated subsets   and   with equal total weights. b) Any subset C of  consisting of at least 7 stones contains two separated subsets   with equal total weights.   Solution Congratulations Jacob Tsimerman Toronto, Canada Julien Santini Universite de Provence, France Ahmet Şensoy Bilkent University, Ankara Athanasios Papaioannou Thessaloniki, Greece Hung Ha Duy Hanoi University of Education, Vietnam

 April 2003 Question : Prove that the sequence ,  n = 1, 2, …. contains all prime numbers except 2 and 3. Solution Congratulations Jacob Tsimerman Toronto, Canada Ali Yıldız Bilkent University, Ankara Vlad Petrescu University of Florida, USA Hung Ha Duy Hanoi University of Education, Vietnam Ahmet Şensoy Bilkent University, Ankara Athanasios Papaioannou Thessaloniki, Greece Ünsal Atasoy Middle East Technical University, Ankara Stojan Trajanovski High School "RJ Korcagin" Skopje, R. Macedonia Umut Işık Bilkent University, Ankara Talat Şenocak Bilkent University, Ankara Julien Santini Universite de Provence, France Birol Bakay Bilkent University, Ankara Zekeriya Yalçın Karataş Middle East Technical University, Ankara Bruno Langlois Lycee Jean Rostand, Mantes, France Özgür Ocak Istanbul Technical University Erdem Özcan Bilkent University, Ankara Çağrı Özçağlar Bilkent University, Ankara Ali İlik , Gürel Yıldız Işık University, Istanbul Şener Öztürk

 May 2003 Question : The polynomial  with nonnegative coefficients has n real roots. Prove that .Solution Congratulations Ahmet Şensoy Bilkent University, Ankara Jacob Tsimerman Toronto, Canada Athanasios Papaioannou Thessaloniki, Greece Atilla Yılmaz Bosporus University, Istanbul Gürel Yıldız Işık University, Istanbul Zekeriya Yalçın Karataş Middle East Technical University, Ankara Ünsal Atasoy Middle East Technical University, Ankara Stojan Trajanovski High School "RJ Korcagin" Skopje, R. Macedonia Hung Ha Duy Hanoi University of Education, Vietnam Yüksel Demir Özel Aziziye Lisesi, Erzurum

 June 2003 Question : Consider two polynomials  and  with integer coefficients. Suppose that for all real values of x  ,  and .  Prove that  for all values of x. .Solution Congratulations Atilla Yılmaz Bosporus University, Istanbul Ali Yıldız Bilkent University, Ankara Stojan Trajanovski High School "RJ Korcagin" Skopje, R. Macedonia Athanasios Papaioannou Thessaloniki, Greece Richard Pinch Cheltenham, United Kingdom

 July-August 2003 Question : Are there two polynomials  and  with real coefficients such that for any integer k,  is integer but ,  and  are not integers? Solution Congratulations Bruno Langlois Lycee Rabelais, Meudon, France Mustafa Turgut Isparta Athanasios Papaioannou Thessaloniki, Greece Fatih Selimefendigil Istanbul Technical University

 September 2003 Question : Solve in natural numbers: Solution Congratulations Henry Shin University of California, San Diego, USA Julien Santini Universite de Provence, France Jacob Tsimerman Toronto, Canada Fatih Selimefendigil Istanbul Technical University Tomas Jurik Comenius University, Bratislava, Slovakia Jan Mazak Comenius University, Bratislava, Slovakia

 October 2003 Question : Let P(x) be a polynomial with integer coefficients: . Assume that the equation  has at least one integer solution for  i = 1, 2, 3. Prove that the equation  has at most one integer solution.   Solution Congratulations Jan Mazak Comenius University, Bratislava, Slovakia Jacob Tsimerman Toronto, Canada Henry Shin University of California, San Diego, USA Ali Adali Bilkent University, Ankara Fatih Selimefendigil Istanbul Technical University Abdullah Turan Istanbul Technical University

 November 2003 Question : Find all pair of natural numbers a and b satisfying the equation : . Solution Congratulations Usko Lahti Hyvinkaan Sveitsin lukio, Finland Athanasios Papaioannou Boston, USA Henry Shin University of California, San Diego, USA Ihsan Aydemir Umraniye Lycee, Istanbul Cihan Okay Bilkent University Andrei Negut Bucharest, Romania Ali Adali Bilkent University Michael Lipnowski St.John's Ravenscourt School, Winnipeg, Canada Jacob Tsimerman Toronto, Canada Birol Yesiltepe Marmara University Bruno Langlois Lycee Rabelais, Meudon, France

 December 2003 Question : Suppose that  ,  for . Prove that one can divide these n numbers  into 11 groups such that the sum of numbers in each group does not exceed 15. Congratulations Usko Lahti Hyvinkaan Sveitsin lukio, Finland Athanasios Papaioannou Boston, USA Ali Adali Bilkent University Michael Lipnowski St.John's Ravenscourt School, Winnipeg, Canada