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Source: 5th German TST 2005, problem 2, not from the shortlist

May 12, 2005
inequalities proposedinequalities

Problem Statement

Let n be a positive integer, and let a1a_1, a2a_2, ..., ana_n, b1b_1, b2b_2, ..., bnb_n be positive real numbers such that a1≥a2≥...≥ana_1\geq a_2\geq ...\geq a_n and b1≥a1b_1\geq a_1, b1b2≥a1a2b_1b_2\geq a_1a_2, b1b2b3≥a1a2a3b_1b_2b_3\geq a_1a_2a_3, ..., b1b2...bn≥a1a2...anb_1b_2...b_n\geq a_1a_2...a_n. Prove that b1+b2+...+bn≥a1+a2+...+anb_1+b_2+...+b_n\geq a_1+a_2+...+a_n.
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