MathDB

Find maximum number of pairs whose product is at least 1

Source: 2024 China MO, Day 2, Problem 4

November 29, 2023

algebra

Problem Statement

Let

a_1, a_2, \ldots, a_{2023}

be nonnegative real numbers such that

a_1 + a_2 + \ldots + a_{2023} = 100

. Let

A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}

. Prove that

|A| \leqslant 5050

and determine when the equality holds.
Proposed by Yunhao Fu