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here is a regular n-gon whose all vertices lie on the considered arcs

Source: Romania IMO TST 1991 p6

February 19, 2020

combinatoricscombinatorial geometry

Problem Statement

Let

n \ge 3

be an integer. A finite number of disjoint arcs with the total sum of length

1 -\frac{1}{n}

are given on a circle of perimeter

1

. Prove that there is a regular

n

-gon whose all vertices lie on the considered arcs