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Found problems: 1

2024 Tuymaada Olympiad, 3
Tags: number theory , estimates
Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the start and finished training. At what maximum $S$ can we knowingly say that at some point the sum of the pairwise distances between athletes was at least $S$? [i]Proposed by A. Golovanov, I. Rubanov[/i]