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\sum a_i^3 \ge t(\sum a_i )^2 when a_i \in N-{0}

Source: Norwegian Mathematical Olympiad 2003 - Abel Competition p2b

February 21, 2020
Suminequalitiesnumber theory

Problem Statement

Let a1,a2,...,ana_1,a_2,...,a_n be nn different positive integers where n≥1n\ge 1. Show that ∑i=1nai3≥(∑i=1nai)2\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2
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