MathDB

A. 852

Part of KoMaL A Problems 2022/2023

Problems(1)

Sharp inequality with strange condition

Source: KoMaL A. 852.

5/11/2023
Let (ai,bi)(a_i,b_i) be pairwise distinct pairs of positive integers for 1≤i≤n1\le i\le n. Prove that (a1+a2+…+an)(b1+b2+…+bn)>29n3,(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>29c>\frac29 it is possible that (a1+a2+…+an)(b1+b2+…+bn)<cn3.(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
Inequalityalgebrainequalities