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.

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 

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).

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 

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?

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 .

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.

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, .

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 . .

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 