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.
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Congratulations |
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Ali Nabi Duman | Bilkent Univ. | ||||||
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Gary C. Alexander | Georgia Inst. of Technology | |||||
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Emre Per | Bilkent Univ. |
March 2001 |
Question :
Find all pairs of nonnegative integer numbers (m,n) satisfying the equality .
Solution
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Congratulations |
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Necdet Batır |
YüzüncüYıl Univ.,
Van |
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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.
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Congratulations |
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Hasip Yılmazoğlu |
Şanlıurfa
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Necdet Batır |
YüzüncüYıl Univ.,
Van
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Özkan Bozdeş |
Bilkent University |
May 2001 |
Question :
Prove that for all
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Congratulations |
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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.
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Congratulations |
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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.
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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
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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
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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.
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Congratulations | ||||
Ahmet Cetintas | Bilkent University | ||||||
Aytek Arikoglu | Bilkent University | ||||||
Murat Ak | Bilkent University |