Let
![\nu](/media/m/2/5/f/25f665896681d1e779f693c550249640.png)
be an irrational positive number, and let
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
be a positive integer. A pair
![(a, b)](/media/m/0/5/9/059101434c13a86d8297575c7ee676ea.png)
of positive integer is called
good if
![a \lceil b \nu \rceil - b \lfloor a \nu \rfloor = m \text{.}](/media/m/3/1/d/31dbc7777c8ac34764b576beee3a3706.png)
A good pair
![(a, b)](/media/m/0/5/9/059101434c13a86d8297575c7ee676ea.png)
is called
excellent if neither of the pairs
![(a - b, b)](/media/m/3/b/1/3b18a79d88693e68d1f24c810292db83.png)
and
![(a, b - a)](/media/m/0/5/b/05b9fe24a1912cb14e27c22272fb9f8d.png)
is good. (As usual, by
![\lfloor x \rfloor](/media/m/c/c/2/cc22bc897f71e3436c8e79a0a632e862.png)
and
![\lceil x \rceil](/media/m/1/c/d/1cde8b0b8970f18c2d13673a4d03d064.png)
we denote the integer numbers such that
![x - 1 < \lfloor x \rfloor \leq x](/media/m/9/7/c/97ced09f26ccc21d14bd503e6cbf369a.png)
and
![x \leq \lceil x \rceil < x + 1](/media/m/e/8/d/e8dd0effd2a17abbb6a0d74f35270e14.png)
.)
Prove that the number of excellent pairs is equal to the sum of the positive divisors of
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
.
%V0
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair $(a, b)$ of positive integer is called [i]good[/i] if $$
a \lceil b \nu \rceil - b \lfloor a \nu \rfloor = m \text{.}
$$
A good pair $(a, b)$ is called [i]excellent[/i] 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 $m$.