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ASU 109 All Soviet Union MO 1968 1, 1/2, ... , 1/n

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June 20, 2019
algebraSequenceinequalities

Problem Statement

Two finite sequences a1,a2,...,an,b1,b2,...,bna_1,a_2,...,a_n,b_1,b_2,...,b_n are just rearranged sequence 1,1/2,...,1/n1, 1/2, ... , 1/n with a1+b1≥a2+b2≥...≥an+bn.a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n. Prove that am+an≥4/ma_m+a_n\ge 4/m for every mm (1≤m≤n1\le m\le n) .
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