2.2. $[2+2]$ On a sheet, the numbers 1, $x$, and $y$ are written. In one move, the reciprocal of any of the already written numbers, or the sum or difference of two of the written numbers can be written on the sheet. Is it possible for the number $x^{2}$ to appear on the sheet after several moves? What about the number $x y$?
Solution. a) See 1.2. b) The given operations allow the recorded number $z$ to be divided by $2: \frac{1}{z}, \frac{1}{z}, \frac{2}{z}, \frac{z}{2}$. From the numbers $x$ and $y$, we get $x+y$ and $x-y$. Then from a) we know that $(x+y)^{2}$ and $(x-y)^{2}$ can also be obtained. By forming the difference $(x+y)^{2}-(x-y)^{2}=4 x y$, it remains to divide by 2 twice.