IMO Shortlist 2013 problem N7

  Avg: 0,0
  Avg: 9,0
Dodao/la: arhiva
21. rujna 2014.
Let \nu be an irrational positive number, and let m be a positive integer. A pair (a, b) of positive integer is called good if 
  a \lceil b \nu \rceil - b \lfloor a \nu \rfloor = m \text{.}

A good pair (a, b) is called excellent if neither of the pairs (a - b, b) and (a, b - a) is good. (As usual, by \lfloor x \rfloor and \lceil x \rceil we denote the integer numbers such that x - 1 < \lfloor x \rfloor \leq x and x \leq \lceil x \rceil < x + 1.)

Prove that the number of excellent pairs is equal to the sum of the positive divisors of m.
Izvor: U.S.A.