MEMO 2015 pojedinačno problem 2


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28. kolovoza 2018.
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Let n \geqslant 3 be an integer. An inner diagonal of a simple n-gon is a diagonal that is contained in the n-gon. Denote by D(P) the number of all inner diagonals of a simple n-gon P and by D(n) the least possible value of D(Q), where Q is a simple n-gon. Prove that no two inner diagonals of P intersect (except possibly at a common endpoint) if and only if D(P) = D(n).

Remark: A simple n-gon is a non-self-intersecting polygon with n vertices. A polygon is not necessarily convex.

Izvor: Srednjoeuropska matematička olimpijada 2015, pojedinačno natjecanje, problem 2