BY-MO 1999 Problem 3

A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$ , and replace them by the numbers $$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right)$$ respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$ , is it possible to obtain a sequence whose all terms are integers?