IMO Shortlist 1982 problem 6

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Dodao/la: arhiva
April 2, 2012
Let S be a square with sides length 100. Let L be a path within S which does not meet itself and which is composed of line segments A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n with A_0=A_n. Suppose that for every point P on the boundary of S there is a point of L at a distance from P no greater than 1\over2. Prove that there are two points X and Y of L such that the distance between X and Y is not greater than 1 and the length of the part of L which lies between X and Y is not smaller than 198.
Source: Međunarodna matematička olimpijada, shortlist 1982