MATH PROBLEMS OF 2003
January 2003 |
Question : Let a1, a2, … , an be n natural numbers satisfying ai ≤ i for each i = 1, … ,n . If is even, prove that at least one expression of the form a1 ± a2 ± a3 ± … ± an is equal to zero. |
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Congratulations | ||||
Julien Santini | Universite de Provence, France | ||||||
Athanasios Papaioannou |
Thessaloniki, Greece |
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Stojan Trajanovski |
High School "RJ Korcagin" Skopje, R. Macedonia |
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Jacob Tsimerman | Toronto, Canada | ||||||
Hung Ha Duy |
Hanoi University of Education, Vietnam |
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Ti Yin | Toronto, Canada | ||||||
February 2003 |
Question : Prove that the equation has infinitely many natural solutions. |
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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 |
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Athanasios Papaioannou |
Thessaloniki, Greece |
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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 |
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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 |
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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.
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Congratulations | ||||
Jacob Tsimerman | Toronto, Canada | ||||||
Julien Santini | Universite de Provence, France | ||||||
Ahmet Şensoy | Bilkent University, Ankara | ||||||
Athanasios Papaioannou |
Thessaloniki, Greece |
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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. |
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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 |
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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
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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 |
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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. |
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Congratulations | ||||
Atilla Yılmaz | Bosporus University, Istanbul | ||||||
Ali Yıldız | Bilkent University, Ankara | ||||||
Stojan Trajanovski |
High School "RJ Korcagin" Skopje, R. Macedonia |
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Athanasios Papaioannou | Thessaloniki, Greece | ||||||
Richard Pinch | Cheltenham, United Kingdom | ||||||
July-August 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:
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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.
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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 | ||||||