IMO 2021 Shortlist N4
Alice is given a rational number r ą 1 and a line with two points B ‰ R, where point R contains a red bead and point B c...
Category: Number Theory
Problem
Alice is given a rational number r ą 1 and a line with two points B ‰ R, where point R contains a red bead and point B contains a blue bead. Alice plays a solitaire game by performinga 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 X1 with ÝÝÑ Y X1 “rk Ý Ý Ñ Y X. Alice’s goal is to move the red bead to the point B. Find all rational numbers r ą 1 such that Alice can reach her goal in at most 2021 moves.