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Turkey NMO 2006 1st Round - P12 (Combinatorics)

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February 2, 2013

Problem Statement

In how many different ways can the set {1,2,…,2006}\{1,2,\dots, 2006\} be divided into three non-empty sets such that no set contains two successive numbers?
<spanclass=′latex−bold′>(A)</span> 32006−3⋅22006+1<spanclass=′latex−bold′>(B)</span> 22005−2<spanclass=′latex−bold′>(C)</span> 32004<spanclass=′latex−bold′>(D)</span> 32005−1<spanclass=′latex−bold′>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 3^{2006}-3\cdot 2^{2006}+1 \qquad<span class='latex-bold'>(B)</span>\ 2^{2005}-2 \qquad<span class='latex-bold'>(C)</span>\ 3^{2004} \qquad<span class='latex-bold'>(D)</span>\ 3^{2005}-1 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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