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.

Solution

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.

.

Solution

 

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.

 

Solution

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 ≤ --1--- V~ 1-x2

is held. Find the maximal possible value of

 a2 + c.

 

Solution

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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