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2016 CNMO Grade 10/11 P1

Source: 2016 China Northern MO Grade 10, Problem 1; 2016 China Northern MO Grade 11, Problem 1

February 24, 2020
algebra

Problem Statement

a1,a2,⋯ ,ana_1,a_2,\cdots,a_n are positive real numbers, a1+a2+⋯ ,an=1a_1+a_2+\cdots,a_n=1. Prove that ∑m=1nam∏k=1m(1+ak)≤1−12n.\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.
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