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Let n\geq 3 be a fixed integer. Each side and each diagonal of a regular n-gon is labelled with a number from the set \left\{1;\;2;\;...;\;r\right\} in a way such that the following two conditions are fulfilled:

1. Each number from the set \left\{1;\;2;\;...;\;r\right\} occurs at least once as a label.

2. In each triangle formed by three vertices of the n-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side.

(a) Find the maximal r for which such a labelling is possible.

(b) Harder version (IMO Shortlist 2005): For this maximal value of r, how many such labellings are there?

Easier version (5th German TST 2006) - contains answer to the harder versionEasier version (5th German TST 2006): Show that, for this maximal value of r, there are exactly \frac{n!\left(n-1\right)!}{2^{n-1}} possible labellings.

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