Problem

Source: 2022 China TST, Test 2, P3

Tags: inequalities, number theory, Divisibility



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.