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Find all positive integers n

Source: 1979 Austrian-Polish Mathematical Competition p3

February 15, 2020
inequalities

Problem Statement

Find all positive integers nn such that the inequality (∑i=1nai2)(∑i=1nai)−∑i=1nai3≥6∏i=1nai\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i holds for any nn positive numbers a1,…,ana_1, \dots, a_n.
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