MEMO 2009 pojedinačno problem 2

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28. travnja 2012.
Suppose that we have n \ge 3 distinct colours. Let f(n) be the greatest integer with the property that every side and every diagonal of a convex polygon with f(n) vertices can be coloured with one of n colours in the following way:

(i) At least two colours are used,

(ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours.

Show that f(n) \le (n-1)^2 with equality for infintely many values of n.
Izvor: Srednjoeuropska matematička olimpijada 2009, pojedinačno natjecanje, problem 2