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.

Solution

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.

Solution

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.

 

Solution

 

 

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).

Solution

 

 

 

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?

Solution

 

 

 

 

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 .

Solution

 

 

 

 

 

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.

Solution

 

 

 

 

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].

Solution

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.

Solution

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, .

Solution

 

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  .

.

Solution

 

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