January 2003

Question :

Let a₁, a₂, … , a_n be n natural numbers satisfying a_i ≤ i for each i = 1, … ,n . If  is even, prove that at least one

 expression of the form a₁ ± a₂ ± a₃ ± … ± a_n is equal to zero.

Solution

Athanasios Papaioannou

Thessaloniki, Greece

High School "RJ Korcagin"

Skopje, R. Macedonia

Hung Ha Duy

Hanoi University of Education, Vietnam

February 2003

Question :

Prove that the equation  has infinitely many natural solutions.

Solution

High School "RJ Korcagin"

Skopje, R. Macedonia

Athanasios Papaioannou

Thessaloniki, Greece

Vejdi Hasanov

Shumen University, Bulgaria

Indian Institute of Technology, Mumbai, India

March 2003

Question :

Consider a collection of 26 stones weighted 1, 2, 3, … , 26 grams. Let us denote it by . Prove that

a)  There is a subset  of , consisting of 6 stones, which does not contain any two separated subsets   and   with equal total weights.

b) Any subset C of  consisting of at least 7 stones contains two separated subsets   with equal total weights.

Solution

Athanasios Papaioannou

Thessaloniki, Greece

April 2003

Question :

Prove that the sequence ,  n = 1, 2, …. contains all prime numbers except 2 and 3.

Solution

High School "RJ Korcagin"

Skopje, R. Macedonia

May 2003

Question :

The polynomial

 with nonnegative coefficients has n real roots. Prove that

.Solution

High School "RJ Korcagin"

Skopje, R. Macedonia

June 2003

Question :

Consider two polynomials  and  with integer coefficients. Suppose that for all real values of x

 ,  and .

 Prove that  for all values of x.

.Solution

High School "RJ Korcagin"

Skopje, R. Macedonia

July-August 2003

Question :

Are there two polynomials  and  with real coefficients such that for any integer k,  is integer but ,  and  are not integers?

Solution

September 2003

Question :

Solve in natural numbers:

Solution

October 2003

Question :

Let P(x) be a polynomial with integer coefficients: . Assume that the equation  has at least one integer solution for

 i = 1, 2, 3. Prove that the equation  has at most one integer solution.

Solution

November 2003

Question :

Find all pair of natural numbers a and b satisfying the equation :

.

Solution

December 2003

Question :

Suppose that  ,  for . Prove that one can divide these n numbers  into 11 groups such that the sum of numbers in each group does not exceed 15.

Solution