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a_{k-1}+a_{k+1} >= 2a_k

Source: Polish MO second round 1967 p1

August 22, 2024
algebrainequalities

Problem Statement

Real numbers a1,a2,...,ana_1,a_2,...,a_n (n≥3n \ge 3) satisfy the conditions a1=an=0a_1 = a_n = 0 and ak−1+ak+1≥2aka_{k-1}+a_{k+1} \ge 2a_k for k=2k = 2,33,...,,...,n−1n -1. Prove that none of the numbers a1a_1,......,ana_n is positive.
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