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Problem 5 IMC 2004 Macedonia

Source:

July 25, 2004
inequalitiesIMCcollege contests

Problem Statement

Let SS be a set of (2nn)+1\displaystyle { 2n \choose n } + 1 real numbers, where nn is an positive integer. Prove that there exists a monotone sequence {ai}1≤i≤n+2⊂S\{a_i\}_{1\leq i \leq n+2} \subset S such that ∣xi+1−x1∣≥2∣xi−x1∣, |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , for all i=2,3,…,n+1i=2,3,\ldots, n+1.
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