KöMaL 2023 A852

Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1 \le i \le n$. Prove that $(a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)>\frac{2}{9} n^3,$ and show that the statement is sharp, i.e. for an arbitrary $c>\frac{2}{9}$ it is possible that $(a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)<cn^3.$