Question 12
Determine if there exists a positive integer pair $(m, n)$, such that
(i) the greatest common divisor of $m$ and $n$ is $1$, and $m \le 2007$,
(ii) for any $k = 1, 2, \dots, 2007$, $\lfloor \frac{nk}{m} \rfloor = \lfloor \sqrt{2}k \rfloor$.
(Here $\lfloor x \rfloor$ stands for the greatest integer less than or equal to $x$.)
(i) the greatest common divisor of $m$ and $n$ is $1$, and $m \le 2007$,
(ii) for any $k = 1, 2, \dots, 2007$, $\lfloor \frac{nk}{m} \rfloor = \lfloor \sqrt{2}k \rfloor$.
(Here $\lfloor x \rfloor$ stands for the greatest integer less than or equal to $x$.)