MEMO 2007 pojedinačno problem 2

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28. travnja 2012.
A set of balls contains n balls which are labeled with numbers 1,2,3,\ldots,n. We are given k > 1 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 k+1 balls with (not necessarily different) labels a_{1},a_{2},\ldots,a_{k+1} satisfying the condition a_{1}+a_{2}+\ldots+a_{k}= a_{k+1}, contains at least one ball of each colour.

Find, depending on k the greatest possible number n which admits such a colouring.
Izvor: Srednjoeuropska matematička olimpijada 2007, pojedinačno natjecanje, problem 2