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"... In 1940 J.F. Pal posed the following problem [2]: Let (X, ρ) be a metric space and let c: [0, 1] → X be a continuous curve and n a positive integer. Is there a partition 0 < s1 < s2 < · · · sn < 1 such that ρ(c(si−1), c(si)) = ρ(c(si), c(si+1)), for i = 1, 2,..., n, where s0 = 0 and ..."

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In 1940 J.F. Pal posed the following problem [2]: Let (X, ρ) be a metric space and let c: [0, 1] → X be a continuous curve and n a positive integer. Is there a partition 0 < s1 < s2 < · · · sn < 1 such that ρ(c(si−1), c(si)) = ρ(c(si), c(si+1)), for i = 1, 2,..., n, where s0 = 0 and sn+1 = 1? One can generalize the problem by taking X to be merely a topological space and ρ: X × X → R+ a continuous function such that ρ(x, y) = 0 if and only if x = y (not necessarily a distance function). Moreover, one can require λi−1ρ(c(si−1), c(si)) = λiρ(c(si), c(si+1)), for i = 1, 2,..., n, where λ0> 0, λ1> 0,..., λn> 0 are given real numbers. Of course the case n = 1 is trivial, because if F: [0, 1] → R is the continuous function F (s) = λ1ρ(c(s), c(1)) − λ0ρ(c(0), c(s)), then, if c(0) 6 = c(1), we have F (0)> 0 and F (1) < 0 and by the Intermediate Value Theorem there exists some 0 < s1 < 1 such that F (s1) = 0. Every point in F−1(0) is a solution to the problem. In 1954 K. Urbanik gave a proof of the existence of a solution to Pal’s problem based on the Brouwer fixed point theorem [6]. Another proof based again on the Brouwer fixed