Abstract
Consider the circle C of length 1 and a circular arc A of length ℓ∈(0,1). It is shown that there exists k=k(ℓ)∈N, and a schedule for k runners along the circle with k constant but distinct positive speeds so that at any time t≥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,…,k, with constant rationally independent (thus distinct) speeds ξ1,ξ2,…,ξk, run clockwise along a circle of length 1, starting from arbitrary points. For every circular arc A⊂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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