MATH PROBLEMS OF 2004
January 2004 |
Question : For each real x, let [x] be the maximal integer not exceeding x. Prove that the sequence n =1,2,3, … contains infinitely many composite numbers. |
Congratulations | |||||
Henry Shin | University of California, San Diego, USA | ||||||
Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
Boris Bukh | University of California, Berkeley, USA | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada | ||||||
Jacob Tsimerman | Toronto, Canada | ||||||
Ali Adali | Bilkent University | ||||||
François Glineur | Mons, Belgium | ||||||
Yufei Zhao | Don Mills C.I., Toronto, Canada |
February 2004 |
Question : Let a and b be two integer numbers, a ≠ b. Prove that the polynomial can not be expressed as a product of two nonconstant polynomials with integer coefficients. |
Congratulations | |||||
Yufei Zhao | Don Mills C.I., Toronto, Canada | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Jacob Tsimerman | Toronto, Canada | ||||||
Ali Adali | Bilkent University, Ankara | ||||||
Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
R. Hood | B.C. Hydro, British Columbia, Canada | ||||||
Umut Uludag | Michigan State University, USA | ||||||
Rusen Kaya | Cukurova University, Adana | ||||||
François Glineur | Mons, Belgium | ||||||
Fatih Selimefendigil | Istanbul Technical University, Istanbul | ||||||
Bezirgen Veliyev | Boğaziçi University, Istanbul | ||||||
Henry Shin | University of California, San Diego, USA | ||||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada |
March 2004 |
Question : Find the number of all pairs (a, b) of natural numbers a and b less than 2004 such that is also a natural number.
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Congratulations | |||||
Fatih Selimefendigil | Istanbul Technical University, Istanbul | ||||||
Yufei Zhao | Don Mills C.I., Toronto, Canada | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
François Glineur | Mons, Belgium | ||||||
Henry Shin | University of California, San Diego, USA | ||||||
Fahri Alkan | Bilkent University, Ankara | ||||||
Emre Cakir | Bilkent University, Ankara | ||||||
Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
John Campbell | John F. Ross, C.V.I., Guelph, Ontorio, Canada | ||||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada | ||||||
Ha Duy Hung |
Hanoi University of Education, Vietnam |
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Ali Adali | Bilkent University, Ankara | ||||||
Vlad Petrescu | University of Florida, USA | ||||||
Dimitri Dziabenko | Don Mills Middle School, Toronto, Canada | ||||||
Ihsan Aydemir | Umraniye Lisesi, Istanbul |
April 2004 |
Question : Let , where . Prove that a is integer and find a (mod 5).
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Congratulations | |||||
Yufei Zhao | Don Mills C.I., Toronto, Canada | ||||||
Fatih Selimefendigil | Istanbul Technical University, Istanbul | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Ali Adali | Bilkent University, Ankara | ||||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada | ||||||
Suat Gumussoy | Ohio State University, USA | ||||||
Ha Duy Hung |
Hanoi University of Education, Vietnam |
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Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
Rusen Kaya | Cukurova University, Adana | ||||||
Luigi Bernardini | Monza, Italy | ||||||
Vlad Petrescu | University of Florida, USA | ||||||
Samet Karakas | Bilkent University, Ankara | ||||||
Onur Erten | Bilkent University, Ankara |
May 2004 |
Question : Is it possible to divide the set of all natural numbers into two groups such that no group contains any infinite arithmetic progression?
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Congratulations | |||||
Henry Shin | University of California, San Diego, USA | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Bruno Langlois | Lycee Rabelais, Meudon, France | ||||||
Luigi Bernardini | Monza, Italy | ||||||
Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
Mustafa Turgut | Isparta | ||||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada | ||||||
Büşra Turgut | Antalya | ||||||
Ali Adalı | Bilkent University, Ankara |
June 2004 |
Question : Let a, b and c be nonnegative numbers satisfying . Prove that .
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Congratulations | |||||
Michael Lipnowski | St.John's Ravenscourt School, Winnipeg, Canada | ||||||
Jan Mazak | Camenius University, Bratislava, Slovakia | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
Julien Santini | Universite de Provence, France | ||||||
Henry Shin | University of California, San Diego, USA | ||||||
Caner Koca | Bilkent University, Ankara | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Fatih Selimefendigil | Technical University of Munich, Germany | ||||||
François Glineur | Mons, Belgium | ||||||
Luigi Bernardini | Monza, Italy | ||||||
Dimitri Dziabenko | Don Mills C.I., Toronto, Canada | ||||||
Usko Lahti | Hyvinkaan Sveitsin lukio, Finland | ||||||
Pisupati Raja Sektar | Madras, India | ||||||
Meagan Thompson | Cambridge, Massachusetts, USA | ||||||
Mehmet Uzunkal | Sabancı University, Istanbul | ||||||
Mehdi Abdeh - Kolahchi | Halifax West High School, Halifax, Canada |
July - August 2004 |
Question : Permute some elements of the natural sequence 1, 2, 3, ... such that after this permutation the sum of first k terms is divisible by k.
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Congratulations | |||||
François Glineur | Mons, Belgium | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Julien Santini | Universite de Provence, France | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
September 2004 |
Question : The function f is defined on [0,1] and satisfies the following conditions a) b) for any Prove that the equation has infinitely many solutions. Give an example of such function which is not identically zero on any subinterval of [0,1]. |
Congratulations | |||||
Özcan Yazıcı | Middle East Technical University, Ankara | ||||||
Henry Shin | University of California, San Diego, USA | ||||||
Sridhar Eaturu | Indian Institute of Technology, Bombay, India | ||||||
Julien Santini | Universite de Provence, France | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
Yunus Karabulut | Boğaziçi University, Istanbul | ||||||
October 2004 |
Question : Prove the inequality for any real x and natural n. |
Congratulations | |||||
Athanasios Papaioannou | Boston, USA | ||||||
François Glineur | Mons, Belgium | ||||||
Asger Hvide Olesen | Toender, Denmark | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
Konstantinos Drakakis | University of Edinburg, UK | ||||||
November 2004 |
Question : Find nonnegative real numbers , , , and such that for each i = 2, 3, 4, 5,6 the inequality , (*) is held. Prove that for arbitrary nonnegative real numbers , , , and the inequality (*) is held for at least one i, .
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Congratulations | |||||
Seymur Cahangirov | Hacettepe University, Ankara | ||||||
Asger Hvide Olesen | Toender, Denmark | ||||||
Konstantinos Drakakis | University of Edinburg, UK | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
Athanasios Papaioannou | Boston, USA | ||||||
Jan Mazak | Camenius University, Bratislava, Slovakia | ||||||
Ekrem Emre | Kutahya | ||||||
Aycan Uslu | Samanyolu Fen Lisesi, Ankara |
December 2004 |
Question : Find all natural numbers a such that at some natural n, first digits of and are a: and . .
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Congratulations | |||||
Athanasios Papaioannou | Boston, USA | ||||||
Ali Adalı | Bilkent University, Ankara | ||||||
G.R.A.20 Math Problems Group | Italy | ||||||
İhsan Yücel | Ondokuz Mayıs University, Samsun | ||||||
Yunus Esencayı | Middle East Technical University, Ankara | ||||||
İsmail Sağlam | Bilkent University, Ankara | ||||||
Samet Karakaş | Bilkent University, Ankara | ||||||
Onur Erten | Bilkent University, Ankara | ||||||
Konstantinos Drakakis | University of Edinburg, UK |