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IMO ShortList 2008, Number Theory problem 2

Source: IMO ShortList 2008, Number Theory problem 2, German TST 2, P2, 2009

July 9, 2009
number theorymodular arithmeticSequenceDivisibilityIMO Shortlist

Problem Statement

Let a1 a_1, a2 a_2, … \ldots, an a_n be distinct positive integers, n≥3 n\ge 3. Prove that there exist distinct indices i i and j j such that a_i \plus{} a_j does not divide any of the numbers 3a1 3a_1, 3a2 3a_2, … \ldots, 3an 3a_n. Proposed by Mohsen Jamaali, Iran
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