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a_1 + 2a_2 + 3a_3 + ... + na_n > 0 if a_1 + a_2 + ... + a_n = 0

Source: 1966 Swedish Mathematical Competition p2

March 21, 2021
algebrainequalities

Problem Statement

a1+a2+...+an=0a_1 + a_2 + ... + a_n = 0, for some kk we have aj≤0a_j \le 0 for j≤kj \le k and aj≥0a_j \ge 0 for j>kj > k. If ai are not all 00, show that a1+2a2+3a3+...+nan>0a_1 + 2a_2 + 3a_3 + ... + na_n > 0.
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