Question 27
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair $(a, b)$ of positive integers is called good if
$$
a\lceil b \nu\rceil-b\lfloor a \nu\rfloor=m
$$
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 $x-1\lt \lfloor x\rfloor \leqslant x$ and $x \leqslant\lceil x\rceil\lt x+1$.)
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
$$
a\lceil b \nu\rceil-b\lfloor a \nu\rfloor=m
$$
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 $x-1\lt \lfloor x\rfloor \leqslant x$ and $x \leqslant\lceil x\rceil\lt x+1$.)
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.