There are several rational numbers written on a board. If the numbers $x{}$ and $y{}$ are present on the board, we can add the number $(x+y)/(1-xy)$ to it. Prove that there is a rational number that cannot ever appear on the board.
Source: 239-School Open Olympiad 2022, Junior League P8
Tags: combinatorics, number theory
There are several rational numbers written on a board. If the numbers $x{}$ and $y{}$ are present on the board, we can add the number $(x+y)/(1-xy)$ to it. Prove that there is a rational number that cannot ever appear on the board.