MATH PROBLEMS OF 2001 February 2001 Question :    Let  f(x) = 5 x13 + 13 x5 + 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. Solution 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   x1 + x2 + x3 + x4 - x1x2 - x1x3 - x1x4 - x2x3 - x2x4 - x3x4 + x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 - x1x2x3x4  if  | xi | 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  x3 + y3 + z3 = 2  has infinitely many integer solutions. Congratulations Ahmet Cetintas Bilkent University Aytek Arikoglu Bilkent University Murat Ak Bilkent University