all the turtles will be at the vertices of some convex polygon.
Source: I Soros Olympiad 1994-95 Ukraine R2 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
June 6, 2024
combinatoricscombinatorial geometry
Problem Statement
Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.