MathDB
IMO Shortlist 2013, Number Theory #7

Source: IMO Shortlist 2013, Number Theory #7

July 10, 2014
algebranumber theoryDivisorsIMO Shortlist

Problem Statement

Let ν\nu be an irrational positive number, and let mm be a positive integer. A pair of (a,b)(a,b) of positive integers is called good if a⌈bν⌉−b⌊aν⌋=m.a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m. A good pair (a,b)(a,b) is called excellent if neither of the pair (a−b,b)(a-b,b) and (a,b−a)(a,b-a) is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of mm.
Back to ProblemsView on AoPS