MATH PROBLEMS OF 2002

January 2002

Question :

Find the minimum of   if   and  .

Solution

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.

Solution

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.

Solution

Congratulations

Baatar Tsolman Middle East Technical University, Ankara

 

June 2002

Question :

Is there an integer n such that  is a rational number?

Solution

Congratulations

 

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

 

Solution

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  

Solution

Congratulations

Ha Duy Hung

Hanoi University of Education, Vietnam

Murat Ak Bilkent University, Ankara
K. Zhereb

Moscow Institute of Physics and Technology, Moscow

Mustafa Turgut Isparta
   
   
   

 

October 2002

Question :

Find all real solutions of the following equation :

Solution

Congratulations

Hung Ha Duy

Hanoi University of Education, Vietnam

Mustafa Turgut

Isparta

Vejdi Hasanov

Shumen University, Bulgaria

Beata Stehlikova Comenius University, Bratislava, Slovakia
Jacob Tsimerman Toronto, Canada
Murat Ak Bilkent University, Ankara
Stojan Trajanovski

High School "RJ Korcagin"

Skopje, R. Macedonia

Eaturu Sribar

Indian Institute of Technology, Mumbai, India

Ignas Buvitis

Lithuania

Athanasios Papaioannou

Thessaloniki, Greece

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 .

Solution

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.

 

Solution

Congratulations
Vivek Kumar Mehra Mumbai, India
Stojan Trajanovski

High School "RJ Korcagin"

Skopje, R. Macedonia

Athanasios Papaioannou

Thessaloniki, Greece

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