MATH PROBLEMS OF 2002
January 2002 |
Question : Find the minimum of if and . |
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Congratulations | ||||
Serhat Doğan | Özel Şehzade Mehmet Lisesi, Manisa | ||||||
Murat Ak | Bilkent Universitesi, Ankara | ||||||
Yankı Lekili | Bilkent Universitesi, Ankara | ||||||
Erkan Özkan | Özel Nilüfer Lisesi, Bursa |
April 2002 |
Question : Prove that the equation
has no solution in natural numbers.
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Congratulations | ||||
Suat Gumussoy | The Ohio State University | ||||||
Murat Ak | Bilkent University, Ankara | ||||||
Baatar Tsolman | Middle East Technical University, Ankara | ||||||
Koksal Dinc | Hacettepe University | ||||||
Erkan Ozkan | Ozel Nilufer Lisesi, Bursa | ||||||
Oztekin Bakir | Bartin |
May 2002 |
Question :
There is a finite number of towns in a country. They are connected by one direction roads. It is known that for any two towns, one of them can be reached from the other one. Prove that there is a town such that all the remaining towns can be reached from it.
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Congratulations |
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Baatar Tsolman | Middle East Technical University, Ankara | ||||||
June 2002 |
Question : Is there an integer n such that is a rational number?
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Congratulations
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Ali Yildiz | Bilkent University, Ankara | ||||||
Vivek Kumar Mehra | Mumbai, India | ||||||
Serhat Dogan | Ozel Sehzade Mehmet Lisesi, Manisa | ||||||
Ha Duy Hung | Hanoi University of Education, Vietnam | ||||||
Sener Ozturk | |||||||
Oztekin Bakir | Bartin | ||||||
Hakan Ozaydin | Middle East Technical University, Ankara | ||||||
Aylin Tokuc | Bilkent University, Ankara | ||||||
Murat Ak | Bilkent University, Ankara |
July-August 2002 |
Question : Non-negative real numbers a, b, and c satisfy . Prove that
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Congratulations | ||||
Vejdi Hasanov | Sumen University, Bulgaria | ||||||
Ha Duy Hung | Hanoi University of Education, Vietnam | ||||||
Murat Ak | Bilkent University, Ankara | ||||||
Vivek Kumar Mehra | Mumbai, India | ||||||
Ali Yıldız | Bilkent University, Ankara | ||||||
Jacob Tsimerman | Toronto, Canada | ||||||
Şener Öztürk | İstanbul | ||||||
Beata Stehlikova | Comenius University, Bratislava, Slovakia |
September 2002 |
Question :
Let x1, x2, … , x2002 be some points lying on a unit circle and dij be the distance between xi and xj . Let be the sum taken over all possible pairs (xi ,xj)
for i, j = 1, 2, … , 2002 and i < j . Find the maximum of S over all possible distributions of x1, x2, … , x2002
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Congratulations | ||||
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Ha Duy Hung |
Hanoi University of Education, Vietnam |
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Murat Ak | Bilkent University, Ankara | ||||||
K. Zhereb |
Moscow Institute of Physics and Technology, Moscow |
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Mustafa Turgut | Isparta | ||||||
October 2002 |
Question : Find all real solutions of the following equation :
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Congratulations | ||||
Hung Ha Duy |
Hanoi University of Education, Vietnam |
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Mustafa Turgut |
Isparta |
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Vejdi Hasanov |
Shumen University, Bulgaria |
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Beata Stehlikova | Comenius University, Bratislava, Slovakia | ||||||
Jacob Tsimerman | Toronto, Canada | ||||||
Murat Ak | Bilkent University, Ankara | ||||||
Stojan Trajanovski |
High School "RJ Korcagin" Skopje, R. Macedonia |
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Eaturu Sribar |
Indian Institute of Technology, Mumbai, India |
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Ignas Buvitis |
Lithuania |
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Athanasios Papaioannou |
Thessaloniki, Greece |
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Şener Öztürk | İstanbul | ||||||
Birol Bakay |
Bilkent University, Ankara |
November 2002 |
Question : Let A be a number obtained by some rearrangement of the digits of 2n, where n is a natural number. Prove that A ≠ 2k for all k > n . |
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Congratulations | ||||
Jacob Tsimerman | Toronto, Canada | ||||||
Beata Stehlikova | Comenius University, Bratislava, Slovakia | ||||||
Birol Bakay | Bilkent University, Ankara | ||||||
Mustafa Öztekin | Boğaziçi University, Istanbul | ||||||
Umut Işık | Bilkent University, Ankara | ||||||
Vivek Kumar Mehra | Mumbai, India | ||||||
Ali Yıldız | Bilkent University, Ankara | ||||||
Yiğit Subaşı | Bilkent University, Ankara | ||||||
Erdem Özcan | Bilkent University, Ankara |
December 2002 |
Question : For each natural number n = p1.p2. … pr , where pi is prime for each i = 1, 2, … , r , define f (n) = 1 + p1 + p2 + … + pr .
Prove that for any natural number k, the sequence a1 = k, am = f(am-1) , m = 2, 3, … is periodic.
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Congratulations | ||||
Vivek Kumar Mehra | Mumbai, India | ||||||
Stojan Trajanovski |
High School "RJ Korcagin" Skopje, R. Macedonia |
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Athanasios Papaioannou |
Thessaloniki, Greece |
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Julien Santini | Universite de Provence, France | ||||||
Emre Çakır | Bilkent University, Ankara | ||||||
Erdem Özcan | Bilkent University, Ankara | ||||||
Umut Işık | Bilkent University, Ankara | ||||||
David Anderson | Middle East Technical University, Ankara | ||||||
Mustafa Öztekin | Boğaziçi University, Istanbul | ||||||
Jacob Tsimerman | Toronto, Canada | ||||||
Beata Stehlikova | Comenius University, Bratislava, Slovakia |