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 |
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. .
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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.
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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.
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |
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Congratulations |
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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 |