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a_1+a_2+...+a_{20}=1518, a_k >= 7k , for k = 1,2,..., 20

Source: Netherlands - Dutch NMO 1963 p3

January 31, 2023
inequalitiesSumalgebra

Problem Statement

Twenty numbers a1,a2,..,a20a_1,a_2,..,a_{20} satisfy: ak≥7k     for     k=1,2,...,20a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20 a1+a2+...+a20=1518a_1+a_2+...+a_{20}=1518 Prove that among the numbers k=1,2,...,20k = 1,2,... ,20 there are no more than seventeen, for which ak≥20k−2k2a_k \ge 20k -2k^2.
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