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Exist k_i for such c_i - IMO LongList 1992 ROM1

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September 2, 2010
inequalitiesalgebraSequenceIMO ShortlistIMO Longlist

Problem Statement

Let c1,⋯ ,cn (n≥2)c_1, \cdots, c_n \ (n \geq 2) be real numbers such that 0≤∑ci≤n0 \leq \sum c_i \leq n. Prove that there exist integers k1,⋯ ,knk_1, \cdots , k_n such that ∑ki=0\sum k_i=0 and 1−n≤ci+nki≤n1-n \leq c_i + nk_i \leq n for every i=1,⋯ ,n.i = 1, \cdots , n.
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