MATH PROBLEMS OF 2005

 January 2005 Question : Find the minimal natural number k with the following property: for any natural number n, the number of divisors of  is not 10. Congratulations Athanasios Papaioannou Boston, USA Ali Adalı Bilkent University, Ankara Tomas Jurik Comenius University, Bratislava, Slovakia

 February 2005 Question : Define a sequence of of natural numbers a1, a2, a3, ... such that a1 is not divisible by 5 and an+1=an+dn, where dn is the last digit of an. Prove that this sequence contains infinitely many terms of the form 2k. . Congratulations Athanasios Papaioannou Boston, USA Ertan Kaya Dove Science Academy, Tulsa, Oklahoma, USA Ali Civril Rensselaer Polytechnic Institute, NY, USA Seymur Cahangirov Hacettepe University, Ankara Erdem Özcan Bilkent University, Ankara Tomas Jurik Comenius University, Bratislava, Slovakia Emrah Paksoy Brandeis University, Waltham, MA, USA Aycan Uslu Samanyolu Fen Lisesi, Ankara Samet Karakas Bilkent University, Ankara Onur Erten Bilkent University, Ankara Yunus Esençayı Middle East Technical University, Ankara Deniz Kutluay Bilkent University, Ankara Ali Adalı Bilkent University, Ankara

 March 2005 Question : Suppose that for all a, b, c, d such that d≥c≥b≥a≥0 the inequality             (a+b+c+d)2 ≥ K b c is held. Find the maximal possible value of K. Congratulations Athanasios Papaioannou Boston, USA Seymur Cahangirov Hacettepe University, Ankara Tomas Jurik Comenius University, Bratislava, Slovakia Burak Yıldız Istanbul Technical University, Istanbul Ertan Kaya Dove Science Academy, Tulsa, Oklahoma, USA Yunus Esençayı Middle East Technical University, Ankara Emrah Paksoy Brandeis University, Waltham, MA, USA Melih Üçer Yüksek Sarıkaya İlköğretim Okulu Onur Erten Bilkent University, Ankara Samet Karakas Bilkent University, Ankara Gürel Yıldız Işık University, Istanbul Görkem Özkaya Boğaziçi University, Istanbul Serkan Sahutoğlu Bilkent University, Ankara Aycan Uslu Samanyolu Fen Lisesi, Ankara Tigran Grigoryan Yerevan State University, Armenia Deniz Kutluay Bilkent University, Ankara Çağrı Özçağlar Bilkent University, Ankara Batuhan Karagöz Ankara Fen Lisesi, Ankara Ali Keskin Middle East Technical University, Ankara

 April 2005 Question : Suppose that, for all -1 < x < 1, the inequality ax2 + bx + c ≤ is held. Find the maximal possible value of   + c. Congratulations Tomas Jurik Comenius University, Bratislava, Slovakia Athanasios Papaioannou Boston, USA Ali Adalı Bilkent University, Ankara Yunus Karabulut Boğaziçi University, Istanbul Mustafa Cihat Demircioğlu Atılım University, Ankara

 May 2005 Solution Congratulations Athanasios Papaioannou Boston, USA Engin Yardimci METU Ihsan Yucel Ondokuz Mayis University Yunus Karabulut Bogazici University Tomas Jurik Comenius University, Bratislava, Slovakia Batuhan Karagoz Ankara Fen Lisesi Fatih Gurses Gebze Institute of Technology Deniz Kutluay Bilkent University Murat Ak Bilkent University Yunus Esencayi METU Alessandro Nicolisi University of Tor Vertaga, Rome Cihan Elmaci Bartin, Endustri Meslek Lisesi Ali Keskin METU Husnu Sincar Istanbul Hatice Agacarasi Dokuz Eylul University Mustafa Ayyuru Bilkent University

 June 2005 Solution Congratulations Tomas Jurik Comenius University, Bratislava, Slovakia Athanasios Papaioannou Boston, USA Batuhan Karagoz Ankara Fen Lisesi Nesim Yigit Bilkent University Deniz Kutluay Bilkent University Yunus Karabulut Bogazici University

 July-August 2005 Solution Congratulations Tomas Jurik Comenius University, Bratislava, Slovakia Roberto Tauraso University of Rome, "Tor Vergata", Italy Nesim Yigit Bilkent University Deniz Kutluay Bilkent University Arne Smeets Leuven, Belgium Engin Yardimci METU Gorkem Ozkaya Bogazici University

 September 2005 Solution Congratulations Arne Smeets University of Leuven Gorkem Ozkaya Bogazici University Sinan Karal Marmara University Fatih Haltas Bilkent University Tomas Jurik Comenius University Batuhan Karagoz Ankara Fen Lisesi Samil Akcagil Balikesir University Minh Quang Cao Vinh Long, Vietnam

 October 2005 Solution Congratulations Tomas Jurik Comenius University Engin Yardimci METU Arne Smeets University of Leuven Osman Berat Okutan Bilkent University Fuad Hamidli METU Mehmet Gokcu Bilkent University Samil Akcagil Balikesir University Fatih Haltas Bilkent University Stanislav Volkov University of Bristol Samet Karakas Bilkent University Ibrahim Yazar Bilkent University Onur Erten Lund University Jonathan Schneider University of Toronto Schools Nesim Yigit Bilkent University Ihsan Yucel Amasya Egitim Fakultesi Ekrem Emre Dumlupinar University, Kutahya Mustafa Turgut Afyon Kocatepe University Michael Druker University of Waterloo Faruk Temur Bilkent University Said Amghibech Quebec, Canada Tolgahan Toy Bilkent University Deniz Kutluay Bilkent University

 November 2005 Solution Congratulations Tomas Jurik Comenius University Osman Berat Okutan Bilkent University Said Amghibech Quebec, Canada Fatih Haltas Bilkent University Fuad Hamidli METU Tolgahan Toy Bilkent University Onur Erten Lund University Faruk Temur Bilkent University Samet Karakas Bilkent University Ekrem Emre Dumlupinar University, Kutahya Frank Meng Burnaby South Secondary School, British Columbia, Canada Nesim Yigit Bilkent University

 December 2005 Solution Congratulations Said Amghibech Quebec, Canada Batuhan Karagoz Ankara Fen Lisesi Tomas Jurik Comenius University Onur Erten Lund University Stanislav Volkov University of Bristol Samet Karakas Bilkent University Atafirat Pir Bilkent University Osman Berat Okutan Bilkent University Metehan Ozsoy Samanyolu Fen Lisesi Deniz Kutluay Bilkent University Samil Akcagil Balikesir University Yunus Karabulut Bogazici University Fuad Hamidli METU Nesim Yigit Bilkent University