CN-TST 2022 Problem 9

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$ , $a_i$ is not divisible by $a_j$ . Show that $$a_1+a_2+\cdots+a_n \ge 1.1n^2-2n.$$

Note: A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.