MathDB

Let

(a_i,b_i)

be pairwise distinct pairs of positive integers for

1\le i\le n

. Prove that

(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,

and show that the statement is sharp, i.e. for an arbitrary

c>\frac29

it is possible that

(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.

Submitted by Péter Pál Pach, Budapest, based on an OKTV problem

Problem Statement