MathDB

1

Part of 2019 European Mathematical Cup

Problems(2)

Marking positive integers with 0,1, or 2

Source: 8th European Mathematical Cup, Junior Category, Q1

12/26/2019
Every positive integer is marked with a number from the set {0,1,2}\{ 0,1,2\}, according to the following rule: if a positive integer k is marked with j, then the integer k+j is marked with 0.\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0. Let SS denote the sum of marks of the first 20192019 positive integers. Determine the maximum possible value of SS.
Proposed by Ivan Novak
number theorycombinatorics
Find x,y that gcd(x+y,mn)>1

Source: 8th European Mathematical Cup, Senior Category, Q1

12/26/2019
For positive integers aa and bb, let M(a,b)M(a,b) denote their greatest common divisor. Determine all pairs of positive integers (m,n)(m,n) such that for any two positive integers xx and yy such that x∣mx\mid m and y∣ny\mid n, M(x+y,mn)>1.M(x+y,mn)>1.
Proposed by Ivan Novak
number theory