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particles moving between ends of a segment

Source: 2016 Argentina OMA Finals L3 p6

January 16, 2023
combinatoricstime

Problem Statement

Let ABAB be a segment of length 11. Several particles start moving simultaneously at constant speeds from AA up toB B. As soon as a particle reaches BB , turns around and goes to AA; when it reaches AA, begins to move again towards B B , and so on indefinitely. Find all rational numbersr>1 r>1 such that there exists an instant tt with the following property: For each n≥1n\ge 1 , if n+1n+1 particles with constant speeds 1,r,r2,…,rn1,r,r^2,…,r^n move as described, at instant tt, they all lie at the same interior point of segment ABAB.
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