MathDB

7.3

Part of MathLinks Contest 7th

Problems(1)

0773

Source:

7/7/2008
Let n n be a positive integer, and let M \equal{} \{1,2,\ldots, 2n\}. Find the minimal positive integer m m, such that no matter how we choose the subsets Ai⊂M A_i \subset M, 1≤i≤m 1\leq i\leq m, with the properties: (1) |A_i\minus{}A_j|\geq 1, for all i≠j i\neq j, (2) \bigcup_{i\equal{}1}^m A_i \equal{} M, we can always find two subsets Ak A_k and Al A_l such that A_k \cup A_l \equal{} M (here ∣X∣ |X| represents the number of elements in the set X X.)