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Part of 2019 Mediterranean Mathematics Olympiad

Problems(1)

sequence inequality wanted, r\cdot m_r+m_s ~\ge~ (r+1)(s-1)

Source: Mediterranean Mathematical Olympiad 2019 P2 MMC

7/21/2019
Let m1<m2<⋯<msm_1<m_2<\cdots<m_s be a sequence of s≥2s\ge2 positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer rr with 1≤r<s1\le r<s, prove that r⋅mr+ms ≥ (r+1)(s−1). r\cdot m_r+m_s ~\ge~ (r+1)(s-1).
(Proposed by Gerhard Woeginger, Austria)
inequalitiesalgebraSequence