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.

Solution

Congratulations

March 2001

Question :

Find all pairs of nonnegative integer numbers (m,n) satisfying the equality

.

Solution

Congratulations

Van

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

Van

Özkan Bozdeş

Bilkent University

May 2001

Question :

Prove that for all

Solution

Congratulations

June - July 2001

Question :

Find the maximum of

x₁ + x₂ + x₃ + x₄ - x₁x₂ - x₁x₃ - x₁x₄ -

x₂x₃ - x₂x₄ - x₃x₄ + x₁x₂x₃ + x₁x₂x₄ +

x₁x₃x₄ + x₂x₃x₄ - x₁x₂x₃x₄

if  | x_i | 1,  i = 1,2,3,4.

Solution

Congratulations

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.

Solution

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

Solution

November 2001

Question :

Let . Prove that

Solution

December 2001

Question :

Prove that the equation

x³ + y³ + z³ = 2

has infinitely many integer solutions.

Solution