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exists i with sum a_i_>0 if a_1 + a_2 + ... + a_n > 0

Source: Chile Finals 2013 L2 p3

October 5, 2022
algebrainequalities

Problem Statement

Given a finite sequence of real numbers a1,a2,...,ana_1,a_2,...,a_n such that a1+a2+...+an>0.a_1 + a_2 + ... + a_n > 0. Prove that there is at least one index i i such that ai>0,ai+ai+1>0,...,ai+ai+1+...+an>0.a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.
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