Question 19
Let $AB$ a line segment of length $1$. Several elementary particles start moving simultaneously at constant speeds from $A$ to $B$. As soon as a particle reaches $B$, it turns around and heads to $A$; when reaching $A$, it starts moving to $B$ again, and so on indefinitely.
Find all rational numbers $r \gt 1$ with the following property: For each $n \ge 1$, if $n+1$ particles with speeds $1, r, r^2, \dots, r^n$ move as described, there is a moment when all particles are at the same interior point of segment $AB$. (Ignore the dimensions of the particles; assume that they can all gather at one point.)
Find all rational numbers $r \gt 1$ with the following property: For each $n \ge 1$, if $n+1$ particles with speeds $1, r, r^2, \dots, r^n$ move as described, there is a moment when all particles are at the same interior point of segment $AB$. (Ignore the dimensions of the particles; assume that they can all gather at one point.)