Question 69
Ellie is given a rational number $r \gt 1$ and a line with two points $B \neq R$; a red bead is placed on $R$ and a blue bead on $B$. Ellie uses these to play a solitaire game. In each turn, she chooses an integer $k$ (not necessarily positive) and one bead to move. If the chosen bead is at position $X$ and the other bead is at position $Y$, Ellie moves the chosen bead to the position $X'$ such that $\overrightarrow{YX'} = r^k \overrightarrow{YX}$. Ellie's goal is to move the red bead onto $B$. Find all rational numbers $r \gt 1$ for which Ellie can achieve her goal in at most 2021 turns.