MATH PROBLEMS OF 2001
February 2001 
Question :
Let f(x) = 5 x^{13} + 13 x^{5} + 9 a x
Find the least positive integer a such that 65 divides f(x) for every integer x.


Congratulations 



Ali Nabi Duman  Bilkent Univ.  

Gary C. Alexander  Georgia Inst. of Technology  

Emre Per  Bilkent Univ. 
March 2001 
Question :
Find all pairs of nonnegative integer numbers (m,n) satisfying the equality .
Solution


Congratulations 


Necdet Batır 
YüzüncüYıl Univ.,
Van 


Emre Per  Bilkent Univ. 
April 2001 
Question :
Let k be a natural number such that k+1 is divisible by 24. Prove that the sum of all divisors of k (which are natural numbers) is divisible by 24.


Congratulations 


Hasip Yılmazoğlu 
Şanlıurfa



Necdet Batır 
YüzüncüYıl Univ.,
Van



Özkan Bozdeş 
Bilkent University 
May 2001 
Question :
Prove that for all


Congratulations 

Hasip Yılmazoğlu  Şanlıurfa  
June  July 2001 
Question :
Find the maximum of
x_{1 }+ x_{2 }+ x_{3 }+ x_{4 } x_{1}x_{2}  x_{1}x_{3}  x_{1}x_{4}  x_{2}x_{3}  x_{2}x_{4}  x_{3}x_{4} + x_{1}x_{2}x_{3 }+ x_{1}x_{2}x_{4} + x_{1}x_{3}x_{4} + x_{2}x_{3}x_{4}  x_{1}x_{2}x_{3}x_{4}
if  x_{i } 1, i = 1,2,3,4.


Congratulations 

Serdar Ipek  Bilkent University  
Emre Per  Bilkent University  
Erdinc Irci  Bilkent University 
August  September 2001 
Question :
Find all integers n such that the set {1,2,3,4, ....,n} can be written as the disjoint union of the subsets A , B , C whose sum of elements are equal.


Congratulations  
Gary C. Alexander  Georgia Inst. of Technology  
Yanki Lekili  Bilkent University  
Ahmet Cetintas  Bilkent University 
October 2001 
Question :
Let p be a prime number exceeding 5. Prove that there exists a number k such that each digit in the decimal representation of pk is 1 : pk = 1111...1


Congratulations  
Kadir Kutlu  YüzüncüYıl Univ., Van  
Gary C. Alexander  Georgia Inst. of Technology  
Ali Civril  Bilkent University  
Ahmet Cetintas  Bilkent University 
November 2001 
Question :
Let . Prove that


Congratulations  
Ahmet Cetintas  Bilkent University  
Murat Ak  Bilkent University  
Ahmet Kerim 
December 2001 
Question :
Prove that the equation
x^{3} + y^{3} + z^{3} = 2
has infinitely many integer solutions.


Congratulations  
Ahmet Cetintas  Bilkent University  
Aytek Arikoglu  Bilkent University  
Murat Ak  Bilkent University 