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arXiv · 1508.07289

Problems on Track Runners

Adrian Dumitrescu·Csaba D. Tóth
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Abstract

Consider the circle $C$ of length 1 and a circular arc $A$ of length $\ell\in (0,1)$. It is shown that there exists $k=k(\ell) \in \mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds so that at any time $t \geq 0$, at least one of the $k$ runners is not in $A$. On the other hand, we show the following: Assume that $k$ runners $1,2,\ldots,k$, with constant rationally independent (thus distinct) speeds $ξ_1,ξ_2,\ldots,ξ_k$, run clockwise along a circle of length $1$, starting from arbitrary points. For every circular arc $A\subset C$ and for every $T>0$, there exists $t>T$ such that all runners are in $A$ at time $t$. Several other problems of a similar nature are investigated.

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BibTeXRIS

Adrian Dumitrescu, Csaba D. Tóth. 2017-11-03. Problems on Track Runners. https://arxiv.org/abs/1508.07289

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