2005 IMO Shortlist S05

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?

<details><summary>Easier version (5th German TST 2006) - contains answer to the harder version</summary>*Easier 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.</details>

*Proposed by Federico Ardila, Colombia*

令 $n\geq 3$ 为固定整数。规则 $n$ 边形的每条边和每条对角线都用集合 $\left\{1;\;2;\;...;\;r\right\}$ 中的数字进行标记，以满足以下两个条件：**1.** 集合 $\left\{1;\;2;\;...;\;r\right\}$ 中的每个数字作为标签至少出现一次。**2.** 在由 $n$ 的三个顶点形成的每个三角形中-gon，其中两条边标有相同的数字，并且该数字大于第三边的标签。**(a)** 找到可能出现此类标签的最大 $r$。**(b)** *更难版本 (IMO Shortlist 2005)：* 对于 $r$ 的最大值，有多少个这样的标签？

<details><summary>简单版本(第 5 届德语 TST 2006) - 包含较难版本的答案</summary>*简单版本(第 5 届德语 TST 2006)：* 显示，对于 $r$ 的最大值，恰好有 $\frac{n!\left(n-1\right)!}{2^{n-1}}$ 可能的标签。</details>

*由哥伦比亚 Federico Ardila 提出*