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a_{i-1} + a_{i+1} < 2a_i

Source: 2003 Cuba MO 2.5

September 15, 2024
algebrainequalities

Problem Statement

Let a1,a2,...,a9a_1, a_2, ..., a_9 be non-negative real numbers such that a1=a9=0a_1 = a_9 = 0 and at least one of the remaining terms is different from 00. a) Prove that for some ii (i=2,...,8(i = 2, ..., 8) ,holds that ai−1+ai+1<2ai.a_{i-1} + a_{i+1} < 2a_i. b) Will the previous statement be true, if we change the number 22 for 1.91.9 in the inequality?
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