NL-NWO 2010 Problem 4

(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8})$ , because $x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$ .

(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.

Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.