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Omni-MATH / Let r>1 be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at 0 and a blue bead at 1. In a move, Alice chooses one of the beads and …

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problem

Let r>1r>1 be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at 00 and a blue bead at 11. In a move, Alice chooses one of the beads and an integer k∈Zk \in \mathbb{Z}. If the chosen bead is at xx, and the other bead is at yy, then the bead at xx is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all rr for which Alice can move the red bead to 11 in at most 20212021 moves.
Plain-text mathematical notation (without MathML)
Let r>1 be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at 0 and a blue bead at 1. In a move, Alice chooses one of the beads and an integer k∈Z. If the chosen bead is at x, and the other bead is at y, then the bead at x is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$.

Find all r for which Alice can move the red bead to 1 in at most 2021 moves.
Original LaTeX notation
Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$.

Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

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Source and history

task_id
da743532-f7ed-517d-88b3-83cbaec2a2d6
task_key
test--da743532-f7ed-517d-88b3-83cbaec2a2d6
task_revision_id
1
config / split
/ test

Official source

initial import