January 2002

Question :

Find the minimum of   if   and  .

Solution

April 2002

Question :

Prove that the equation

has no solution in natural numbers.

Solution

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

June 2002

Question :

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

Solution

Congratulations

July-August 2002

Question :

Non-negative real numbers a, b, and c satisfy  . Prove that

Solution

September 2002

Question :

Let x₁, x₂, … , x₂₀₀₂  be some points lying on a unit circle and d_ij be the distance between x_i and x_j . Let  be the sum taken over all possible pairs (x_i ,x_j)

     for    i, j = 1, 2, … , 2002 and   i < j .

 Find the maximum of S over all possible distributions of x₁, x₂, … , x₂₀₀₂  

Solution

Ha Duy Hung

Hanoi University of Education, Vietnam

Moscow Institute of Physics and Technology, Moscow

October 2002

Question :

Find all real solutions of the following equation :

Solution

Hung Ha Duy

Hanoi University of Education, Vietnam

Mustafa Turgut

Isparta

Vejdi Hasanov

Shumen University, Bulgaria

High School "RJ Korcagin"

Skopje, R. Macedonia

Indian Institute of Technology, Mumbai, India

Ignas Buvitis

Lithuania

Athanasios Papaioannou

Thessaloniki, Greece

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 ≠ 2^k for all k > n .

Solution

December 2002

Question :

For each natural number n = p₁.p₂. … p_r , where p_i is prime for

each i = 1, 2, … , r ,

define f (n) = 1 + p₁ + p₂ + … + p_r .

Prove that for any natural number k, the sequence a₁ = k, a_m = f(a_m-1) ,

m = 2, 3, … is periodic.

Solution

High School "RJ Korcagin"

Skopje, R. Macedonia

Athanasios Papaioannou

Thessaloniki, Greece