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Omni-MATH / Let n≥3 be a fixed integer. Each side and each diagonal of a regular n-gon is labelled with a number from the set {1; 2; ...; r} in a way such that the following two conditions are…

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Let n≥3n\geq 3 be a fixed integer. Each side and each diagonal of a regular nn-gon is labelled with a number from the set {1;2;...;r}\left\{1;\;2;\;...;\;r\right\} in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set {1;2;...;r}\left\{1;\;2;\;...;\;r\right\} occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the nn-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal rr for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of rr, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of rr, there are exactly n!(n−1)!2n−1\frac{n!\left(n-1\right)!}{2^{n-1}} possible labellings.[/hide] [i]
Plain-text mathematical notation (without MathML)
Let n≥3 be a fixed integer. Each side and each diagonal of a regular n-gon is labelled with a number from the set {1; 2; ...; r} in a way such that the following two conditions are fulfilled:

[b]1.[/b] Each number from the set {1; 2; ...; r} occurs at least once as a label.

[b]2.[/b] 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.

[b](a)[/b] Find the maximal r for which such a labelling is possible.

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

[hide="Easier version (5th German TST 2006) - contains answer to the harder version"]
[i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of r, there are exactly (n!(n−1)!)/(2^(n−1)) possible labellings.[/hide]

[i]
Original LaTeX notation
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:

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

[b]2.[/b] 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.

[b](a)[/b] Find the maximal $r$ for which such a labelling is possible.

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

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

[i]

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Source and history

task_id
fe7ea94e-744e-52ad-b74f-2894c7531637
task_key
test--fe7ea94e-744e-52ad-b74f-2894c7531637
task_revision_id
1
config / split
/ test

Official source

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