MathDB

2

Part of 2001 Tuymaada Olympiad

Problems(3)

Diophantine equation : (a^2,b^2)+(a,bc)+(b,ac)+(c,ab)=199.

Source: Tuymaada 2001, day 1, problem 2.

4/30/2007
Solve the equation (a2,b2)+(a,bc)+(b,ac)+(c,ab)=199.(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199. in positive integers. (Here (x,y)(x,y) denotes the greatest common divisor of xx and yy.)
Proposed by S. Berlov
greatest common divisornumber theory proposednumber theory
An arrangement of non-zero numbers in a square.

Source: Tuymaada 2001, day 2, problem 2.

4/30/2007
Non-zero numbers are arranged in n×nn \times n square (n>2n>2). Every number is exactly kk times less than the sum of all the other numbers in the same cross (i.e., 2n−22n-2 numbers written in the same row or column with this number). Find all possible kk.
Proposed by D. Rostovsky, A. Khrabrov, S. Berlov
combinatorics proposedcombinatorics
integers in an inifite chessboard so that 101 divides any sum of 10 numbers

Source: Tuymaada Junior 2001 p2

4/30/2019
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any 1010 numbers in a row vertically or horizontal, would be divisible by 101101?
infinite chessboardnumber theorySumdivisible