TR-MO 2022 Problem 3

Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs $(i, j)$ , $1\leq i,j\leq 2022$ , satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$