TUYMAADA 2001 Problem 7

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$ , that $x-\{x\}$ is an integer.)