KöMaL 2016 A665

Let $a_1,a_2,\ldots,a_n$ be distinct positive integers. Show that $3\sum_{i=1}^{n}a_i^5+\bigg(\sum_{i=1}^{n}a_i\bigg)^{2} \ge 4\bigg(\sum_{i=1}^{n}a_i^3\bigg)\bigg(\sum_{i=1}^{n}a_i\bigg).$