MathDB
'n' points on a circle, with 'intervals' and 'sub-intervals'

Source: Iran TST 2011 - Day 1 - Problem 3

May 10, 2011
combinatorics unsolvedcombinatorics

Problem Statement

There are nn points on a circle (n>1n>1). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals FF such that for every element of FF like AA there is almost one other element of FF like BB such that A⊆BA \subseteq B (in this case we call AA is sub-interval of BB). We call an interval maximal if it is not a sub-interval of any other interval. If mm is the number of maximal elements of FF and aa is number of non-maximal elements of F,F, prove that n≥m+a2.n\geq m+\frac a2.
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