MATH PROBLEMS OF 2003
January 2003 
Question : Let a_{1}, a_{2}, … , a_{n} be n natural numbers satisfying a_{i} ≤ i for each i = 1, … ,n . If is even, prove that at least one expression of the form a_{1 }± a_{2} ± a_{3} ± … ± a_{n} is equal to zero. 

Congratulations  
Julien Santini  Universite de Provence, France  
Athanasios Papaioannou 
Thessaloniki, Greece 

Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Jacob Tsimerman  Toronto, Canada  
Hung Ha Duy 
Hanoi University of Education, Vietnam 

Ti Yin  Toronto, Canada  
February 2003 
Question : Prove that the equation has infinitely many natural solutions. 

Congratulations  
Hung Ha Duy  Hanoi University of Education, Vietnam  
Umut Işık  Bilkent University, Ankara  
Jacob Tsimerman  Toronto, Canada  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Athanasios Papaioannou 
Thessaloniki, Greece 

Beata Stehlikova  Comenius University, Bratislava, Slovakia  
Janos Kramar  Toronto, Canada  
Erdem Özcan  Bilkent University, Ankara  
Erkan Melih Şensoy  TED Zonguldak College  
Julien Santini  Universite de Provence, France  
Ahmet Şensoy  Bilkent University, Ankara  
Vejdi Hasanov 
Shumen University, Bulgaria 

Mustafa Öztekin  Bosporus University, Istanbul  
Richard Pinch  Cheltenham, United Kingdom  
Öztekin Bakır  Middle East Technical University, Ankara  
Eaturu Sribar 
Indian Institute of Technology, Mumbai, India 

Eren Çetin  Hacettepe University, Ankara  
Ali Yıldız  Bilkent University, Ankara 
March 2003 
Question : Consider a collection of 26 stones weighted 1, 2, 3, … , 26 grams. Let us denote it by . Prove that a) There is a subset of , consisting of 6 stones, which does not contain any two separated subsets and with equal total weights. b) Any subset C of consisting of at least 7 stones contains two separated subsets with equal total weights.


Congratulations  
Jacob Tsimerman  Toronto, Canada  
Julien Santini  Universite de Provence, France  
Ahmet Şensoy  Bilkent University, Ankara  
Athanasios Papaioannou 
Thessaloniki, Greece 

Hung Ha Duy  Hanoi University of Education, Vietnam  
April 2003 
Question : Prove that the sequence , n = 1, 2, …. contains all prime numbers except 2 and 3. 

Congratulations  
Jacob Tsimerman  Toronto, Canada  
Ali Yıldız  Bilkent University, Ankara  
Vlad Petrescu  University of Florida, USA  
Hung Ha Duy  Hanoi University of Education, Vietnam  
Ahmet Şensoy  Bilkent University, Ankara  
Athanasios Papaioannou  Thessaloniki, Greece  
Ünsal Atasoy  Middle East Technical University, Ankara  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Umut Işık  Bilkent University, Ankara  
Talat Şenocak  Bilkent University, Ankara  
Julien Santini  Universite de Provence, France  
Birol Bakay  Bilkent University, Ankara  
Zekeriya Yalçın Karataş  Middle East Technical University, Ankara  
Bruno Langlois  Lycee Jean Rostand, Mantes, France  
Özgür Ocak  Istanbul Technical University  
Erdem Özcan  Bilkent University, Ankara  
Çağrı Özçağlar  Bilkent University, Ankara  
Ali İlik , Gürel Yıldız  Işık University, Istanbul  
Şener Öztürk 
May 2003 
Question : The polynomial
with nonnegative coefficients has n real roots. Prove that


Congratulations  
Ahmet Şensoy  Bilkent University, Ankara  
Jacob Tsimerman  Toronto, Canada  
Athanasios Papaioannou  Thessaloniki, Greece  
Atilla Yılmaz  Bosporus University, Istanbul  
Gürel Yıldız  Işık University, Istanbul  
Zekeriya Yalçın Karataş  Middle East Technical University, Ankara  
Ünsal Atasoy  Middle East Technical University, Ankara  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Hung Ha Duy  Hanoi University of Education, Vietnam  
Yüksel Demir  Özel Aziziye Lisesi, Erzurum 
June 2003 
Question : Consider two polynomials and with integer coefficients. Suppose that for all real values of x , and . Prove that for all values of x. 

Congratulations  
Atilla Yılmaz  Bosporus University, Istanbul  
Ali Yıldız  Bilkent University, Ankara  
Stojan Trajanovski 
High School "RJ Korcagin" Skopje, R. Macedonia 

Athanasios Papaioannou  Thessaloniki, Greece  
Richard Pinch  Cheltenham, United Kingdom  
JulyAugust 2003 
Question : Are there two polynomials and with real coefficients such that for any integer k, is integer but , and are not integers? 
Congratulations  
Bruno Langlois  Lycee Rabelais, Meudon, France  
Mustafa Turgut  Isparta  
Athanasios Papaioannou  Thessaloniki, Greece  
Fatih Selimefendigil  Istanbul Technical University  
September 2003 
Question : Solve in natural numbers:

Congratulations  
Henry Shin  University of California, San Diego, USA  
Julien Santini  Universite de Provence, France  
Jacob Tsimerman  Toronto, Canada  
Fatih Selimefendigil  Istanbul Technical University  
Tomas Jurik  Comenius University, Bratislava, Slovakia  
Jan Mazak  Comenius University, Bratislava, Slovakia 
October 2003 
Question : Let P(x) be a polynomial with integer coefficients: . Assume that the equation has at least one integer solution for i = 1, 2, 3. Prove that the equation has at most one integer solution.

Congratulations  
Jan Mazak  Comenius University, Bratislava, Slovakia  
Jacob Tsimerman  Toronto, Canada  
Henry Shin  University of California, San Diego, USA  
Ali Adali  Bilkent University, Ankara  
Fatih Selimefendigil  Istanbul Technical University  
Abdullah Turan  Istanbul Technical University 
November 2003 
Question : Find all pair of natural numbers a and b satisfying the equation : . 
Congratulations  
Usko Lahti  Hyvinkaan Sveitsin lukio, Finland  
Athanasios Papaioannou  Boston, USA  
Henry Shin  University of California, San Diego, USA  
Ihsan Aydemir  Umraniye Lycee, Istanbul  
Cihan Okay  Bilkent University  
Andrei Negut  Bucharest, Romania  
Ali Adali  Bilkent University  
Michael Lipnowski  St.John's Ravenscourt School, Winnipeg, Canada  
Jacob Tsimerman  Toronto, Canada  
Birol Yesiltepe  Marmara University  
Bruno Langlois  Lycee Rabelais, Meudon, France 
December 2003 
Question : Suppose that , for . Prove that one can divide these n numbers into 11 groups such that the sum of numbers in each group does not exceed 15. 
Congratulations  
Usko Lahti  Hyvinkaan Sveitsin lukio, Finland  
Athanasios Papaioannou  Boston, USA  
Ali Adali  Bilkent University  
Michael Lipnowski  St.John's Ravenscourt School, Winnipeg, Canada  