Question 135
a) Prove that the equality $\{x\} + \{2y\} = \{y\} + \{2x\}$ for real numbers $x$ and $y$ implies $\{x\} = \{y\}$.
b) Do there exist integers $n \ge 3$ such that the equality $\{x\} + \{ny\} = \{y\} + \{nx\}$ for real numbers $x$ and $y$ implies $\{x\} = \{y\}$?
(Here $\{a\} = a - [a]$, where $[a]$ stands for the greatest integer that does not exceed the real number $a$.)
b) Do there exist integers $n \ge 3$ such that the equality $\{x\} + \{ny\} = \{y\} + \{nx\}$ for real numbers $x$ and $y$ implies $\{x\} = \{y\}$?
(Here $\{a\} = a - [a]$, where $[a]$ stands for the greatest integer that does not exceed the real number $a$.)