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Show that there exists a subset of {1, 2, ..., n}

Source: Iran Third Round MO 1998, Exam 4, P4

July 1, 2012
combinatoricscombinatorics proposed

Problem Statement

Let be given r1,r2,…,rn∈Rr_1,r_2,\ldots,r_n \in \mathbb R. Show that there exists a subset II of {1,2,…,n}\{1,2,\ldots,n \} which which has one or two elements in common with the sets {i,i+1,i+2},(1≤i≤n−2)\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2) such that \left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.
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