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A connected set of 4 positive integers

Source: Romanian District Olympiad 2006, Grade 7, Problem 3

March 11, 2006
floor functionfunction

Problem Statement

A set MM of positive integers is called connected if for any element x∈Mx\in M at least one of the numbers x−1,x+1x-1,x+1 is in MM. Let UnU_n be the number of the connected subsets of {1,2,…,n}\{1,2,\ldots,n\}. a) Compute U7U_7; b) Find the smallest number nn such that Un≥2006U_n \geq 2006.
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