Question 71
Alice is given a rational number $r\gt 1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is placed at point $X$, and the other bead is placed at $Y$, then Alice moves the chosen bead to point $X^{\prime}$ with $\overrightarrow{Y X^{\prime}}=r^{k} \overrightarrow{Y X}$.
Alice's goal is to move the red bead to the point $B$. Find all rational numbers $r\gt 1$ such that Alice can reach her goal in at most 2021 moves.
Alice's goal is to move the red bead to the point $B$. Find all rational numbers $r\gt 1$ such that Alice can reach her goal in at most 2021 moves.