IMO Shortlist

Consider the setup of Alice's solitaire game on the number line. Initially, there is a red bead at position $0$ and a blue bead at position $1$. During each move, Alice chooses an integer $k$ and a bead to move. If the red bead is at position $x$ and the blue bead at position $y$, the chosen bead at $x$ will be moved to the new position $x'$ such that:

$$x' - y = r^k (x - y).$$

Initially, the red bead is at $x = 0$ and the blue bead at $y = 1$, hence the difference is $x - y = -1$.

The objective is to determine all values of $r$ for which the red bead can be moved to $1$ in at most 2021 moves.

### Analysis

The condition for moving the red bead from 0 to 1 can be expressed mathematically. We want the final position of the red bead, after a series of moves, to be equal to the position of the blue bead, $y = 1$.

1. Expression of Move:
$x' - y = r^k (x - y)$. Hence, the red bead's new position $x'$ is:
$$x' = y + r^k (x - y).$$
For the red bead (initially at 0) to reach 1, we need:
$$x' = 1.$$
Substituting the initial position ($x = 0$, $y = 1$), the condition becomes:
$$1 = 1 + r^k(-1),$$
$$0 = -r^k,$$
which is not feasible. So what we need instead is:
$$r^k = 1.$$

2. **Possible Values of $r$**:
For $r^k = 1$, $r$ should be such that this product yields a neutral condition after a sequence of operations. Observing that $r > 1$ is bounded rationally, we note that:

- Geometric Interpretation:
Moving the red bead means repeatedly scaling the distance between the beads by $r^k$. Achieving $r^k = \frac{a+1}{a}$ where $a$ is an integer allows the conditions to be met such that one can achieve the bead's movement to completely cover the original difference-scaled path within a limited number of moves.

- Given the constraint that Alice has 2021 moves, exploring the maximum limit of moves to perform reveals that $a$ must be such that $a \leq 1010$ to satisfy the constraints with $2021$ discrete increments potentially covering entire path.

3. Conclusion:
Therefore, for all $r$ values described by the fraction:
$$r = \frac{a+1}{a}$$
where $a \leq 1010$, the condition is met within the given move constraints.

$$\boxed{\left\{ r = \frac{a+1}{a}, a \leq 1010 \right\}}$$
This characterization allows transitioning the red bead from 0 to 1 in precisely $\leq 2021$ moves for the specified values of $r$.