MEMO 2007 pojedinačno problem 2
Kvaliteta:
Avg: 3,6Težina:
Avg: 6,0 A set of balls contains
balls which are labeled with numbers
. We are given
such sets. We want to colour the balls with two colours, black and white in such a way, that
(a) the balls labeled with the same number are of the same colour,
(b) any subset of
balls with (not necessarily different) labels
satisfying the condition
, contains at least one ball of each colour.
Find, depending on
the greatest possible number
which admits such a colouring.
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![1,2,3,\ldots,n](/media/m/c/4/b/c4bd1c039689e9d0b999825f25026eba.png)
![k > 1](/media/m/e/a/0/ea0d9143b264f0f11e7f6efe66f19a42.png)
(a) the balls labeled with the same number are of the same colour,
(b) any subset of
![k+1](/media/m/9/8/7/98741df0d7cec256bef418eea2665342.png)
![a_{1},a_{2},\ldots,a_{k+1}](/media/m/5/4/6/5460e14a7b1692585514e8fca4be3ceb.png)
![a_{1}+a_{2}+\ldots+a_{k}= a_{k+1}](/media/m/7/6/9/769f134260d978399b9d81c265b7932b.png)
Find, depending on
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
Izvor: Srednjoeuropska matematička olimpijada 2007, pojedinačno natjecanje, problem 2