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max # of pairs satisying a^2+b>=1/2021

Source: Turkey National Mathematical Olympiad 2022 P3

December 23, 2022
inequalities

Problem Statement

Let a1,a2,⋯ ,a2022a_1, a_2, \cdots, a_{2022} be nonnegative real numbers such that a1+a2+⋯+a2022=1a_1+a_2+\cdots +a_{2022}=1. Find the maximum number of ordered pairs (i,j)(i, j), 1≤i,j≤20221\leq i,j\leq 2022, satisfying ai2+aj≥12021.a_i^2+a_j\ge \frac 1{2021}.
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