IMO Shortlist 2005 problem C4
Kvaliteta:
Avg: 0,0Težina:
Avg: 7,0 Let
be a fixed integer. Each side and each diagonal of a regular
-gon is labelled with a number from the set
in a way such that the following two conditions are fulfilled:
1. Each number from the set
occurs at least once as a label.
2. In each triangle formed by three vertices of the
-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
for which such a labelling is possible.
(b) Harder version (IMO Shortlist 2005): For this maximal value of
, 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
, there are exactly
possible labellings.
![n\geq 3](/media/m/d/f/e/dfe037b8debb8aa67d6ed7ad5e28cc6c.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![\left\{1;\;2;\;...;\;r\right\}](/media/m/c/a/3/ca3b7606c47708a8f2348f95ac1f8ea3.png)
1. Each number from the set
![\left\{1;\;2;\;...;\;r\right\}](/media/m/c/a/3/ca3b7606c47708a8f2348f95ac1f8ea3.png)
2. In each triangle formed by three vertices of the
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(a) Find the maximal
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
(b) Harder version (IMO Shortlist 2005): For this maximal value of
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
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](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
![\frac{n!\left(n-1\right)!}{2^{n-1}}](/media/m/9/d/e/9ded84c7b4f987a37f450eb6251beafd.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2005