MathDB
2013 Mexican MO, Problem 5

Source:

December 15, 2013
inequalitiesnumber theory unsolvednumber theory

Problem Statement

A pair of integers is special if it is of the form (n,n−1)(n, n-1) or (n−1,n)(n-1, n) for some positive integer nn. Let nn and mm be positive integers such that pair (n,m)(n, m) is not special. Show (n,m)(n, m) can be expressed as a sum of two or more different special pairs if and only if nn and mm satisfy the inequality n+m≥(n−m)2 n+m\geq (n-m)^2 . Note: The sum of two pairs is defined as (a,b)+(c,d)=(a+c,b+d) (a, b)+(c, d) = (a+c, b+d) .
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